Elliot Benjamin Book Collection

NOTE:  This article was published in the Journal of Number Theory, 2015, Issue 154, pp. 118-143. 

 

On the 2-Class Field Tower Conjecture for Imaginary Quadratic Number Fields

with 2-Class group of Rank 4

                                                                              by Elliot Benjamin, Ph.D.     February, 2015

 

Abstract  

     We demonstrate the existence of infinitely many new imaginary quadratic number fields k with 2-class group C_k,2of rank 4 such that k has infinite 2-class field tower. In particular, we demonstrate the existence of new fields k as above when the 4-rank of the class group C_k is equal to 1 or 2, and infinitely many new fields k in the case that the 4-rank of C_k is equal to 1, exactly three negative prime discriminants divide the discriminant d­_k of k, and d_k is not congruent to 4 mod 8. This lends support to the conjecture that all imaginary quadratic number fields k with C_k,2of rank 4 have infinite 2-class field tower.

 

Introduction

     Given an algebraic number field k, we denote by k¹ the Hilbert 2-class field of k, i.e. the maximal abelian unramified extension of k with degree a power of 2. For nonnegative integers n we define the Hilbert 2-class field kⁿ inductively as k⁰ = k and kⁿ⁺¹ = (kⁿ)¹. Denoting by С the containment symbol, we define k⁰ Сk¹ Сk² С … kⁿ С … to be the 2-class field tower of k.

We say that the tower is finite if kⁿ = kⁿ⁺¹ for some n, with length n if n is minimal, and infinite otherwise.

     It is well known that if k is an imaginary quadratic number field, C_k,2 is the 2-Sylow subgroup

of the class group C_k (in the wide sense) of k, and rank C_k,2 is the dimension over F₂ of C­_k/C­_k² where F₂ is the finite field with two elements, then if rank C_k,2 is greater than or equal to 5,    

then k has infinite 2-class field tower [5]. It is also well known that for k imaginary quadratic with rank C_k,2 = 2 or 3, then the 2-class field tower of k may be finite or infinite, and that if            rank C_k,2 = 1 then the 2-class field tower of k is finite and has length 1(cf. [9], [11], [12], [17]). It was conjectured in the late 1970s that if k is imaginary quadratic with rank C_k,2 = 4, then k has infinite 2-class field tower (cf. [12], [13]). In 1996 and 2000 Hajir, extending previous work that Koch (1969) had done in a particular case, used simpler techniques than Koch to prove a partial result in this direction: namely that if k is an imaginary quadratic number field such that C_k has 4-rank of 3 or 4 then k has infinite 2-class field tower (cf. [6], [7] and Lemma 1 below). From our own work in the early 2000s we obtained further partial results in the direction of the 2-class field tower conjecture, focusing upon the number of negative prime discriminants dividing the discriminant d_k of k, the Kronecker symbols of the primes dividing d_k, the congruence class mod 8 of d_k, and the 4-rank of C_k (cf. [1], [2]). Subsequent to our own work, in the 2000s Sueyoshi used Rédei matrices to improve upon our results in the cases when exactly one negative prime discriminant divides d_k and when exactly three and five negative prime discriminants divide d_k, in particular proving that k has infinite 2-class field tower when k is such that the 4-rank of C_k is equal to 1, five negative prime discriminants divide d_k, and d_k is not congruent to 4 mod 8 (cf. [15], [16], [18], [19], [20]). And in 2010 Mouhib improved upon Sueyoshi’s results in the one negative prime discriminant case, proving that k always has infinite 2-class field tower if exactly one negative prime discriminant divides d_k (cf. Lemma 3 and [14]).

     From the above we see that for the cases when exactly three or five negative prime discriminants divide d_k,we, as well as Sueyoshi, have obtained partial results in the direction of the 2-class field tower conjecture (cf. [1], [2], [18], [19], [20] and Lemmas 4 through 7 below).

We summarize the above historical results for when we know the 2-class field conjecture is

satisfied as follows, where k is an imaginary quadratic number field with rank C_k,2 = 4,           “Negative Prime Discriminants” denotes the exact number of negative prime discriminants dividing d_k, “Congruency” denotes whether or not d_k is congruent to 4 mod 8, and “Literature” refers to the researcher who initially completely proved the final result.                        

 

Table 1: Cases when 2-Class Field Conjecture is Satisfied

4-rank of C­_k        Negative Prime Discrminants           Congruency                       Literature                            

3 or 4                               1, 3, or 5                              no restrictions                 Hajir, 1996, 2000

0, 1, 2, 3, or 4                         1                                    no restrictions                 Mouhib, 2010

       2                                     5                                    no restrictions                Benjamin, 2002

       2                                     3                                      not 4 mod 8                     Benjamin, 2002

       1                                     5                                     not 4 mod 8                    Sueyoshi, 2009

    For our present purpose we define “new” imaginary quadratic number fields k with            rank C_k,2 = 4 and infinite 2-class field tower, as fields that do not satisfy the conditions of any of the cases of Lemmas 4 through 7 below, and to the best of our knowledge have not been reported in the literature, either as a part of families or as separate examples.   By applying a result by Schmithals (1980), we are able to demonstrate the existence of new imaginary quadratic number fields k with infinite 2-class field tower when rank C_k,2 = 4 and C_k has 4-rank 1 or 4-rank 2.

In particular, we show that there exist infinitely many new imaginary quadratic number fields k with infinite 2-class field tower when k is such that the 4-rank of C_k is equal to 1, exactly

three negative prime discriminants divide d_k, and d_k is not congruent to 4 mod 8. In the 4-rank 2 case, we obtain new fields k such that exactly three negative prime discriminants divide d_k,

d_k is congruent to 4 mod 8, and the Kronecker symbols of the primes dividing d_k satisfy the conditions we have previously given for which we do not know in general if the 2-class field tower is finite or infinite (cf. [2] and Case C of Lemma 5 below).

 

Preliminaries

     We begin by stating the Golod & Shafarevich Inequality (as refined by Gaschutz and Vinberg), and two related inequalities that have been derived from a more generic inequality by

Martinet (as described in [19]), all of which have been quite useful in obtaining the results that

we give in Lemmas 3 through 7 (cf. [5], [10], [13], [19], [20]).

Lemma 1: Golod & Shafarevich Inequality (refined by Gaschutz and Vinberg): Let k be a number field, C_k be the class group of k, and E_k be the group of units of k. Then the 2-class field tower of k is infinite if rank C_k,2 is greater than or equal to 2 + 2(√(rank₂(E_k) + 1)), where rank₂(E_k) is the dimension of the elementary 2-group E_k/E_k²considered as a vector space over F₂ (and can be described as the number of infinite primes of k).

Lemma 2: i) Let F be a totally real number field of degree n, and E be a totally imaginary quadratic extension of F. Let t be the number of prime ideals of F which ramify in E.                If t ≥ 3 + 2√(n + 1), then the 2-class field tower of E is infinite.

               ii) Let F be a totally imaginary number field of degree n, and E be a quadratic extension of F. Let t be the number of prime ideals of F which ramify in E.                                 If t ≥ (n/2) + 3 + 2(√(n + 1)), then the 2-class field tower of E is infinite.

     We next combine our aforementioned results by Hajir and Mouhib (cf. [5], [6], [14]) into the following lemma.

Lemma 3: Let k be an imaginary quadratic number field such that C_k has 4-rank ≥ 3, or

rank C_k,2 = 4 and exactly one negative prime discriminant divides d_k. Then k has infinite

2-class field tower.

     For the remaining cases when exactly three or five negative prime discriminants divide d_k

for k an imaginary quadratic number field with rank C_k,2 = 4, we state our previous results

(cf. Lemmas 4, 5, and 6), as well as the relevant results of Sueyoshi (cf. Lemma 7), in the form

of the four lemmas given below (cf. [1], [2], [19], [20]). In Lemma 7 we utilize only parts ii)

and iv) in Sueyoshi’s Proposition in [19] and [20], as parts i) and iii) are contained in our own previous results (see cases A and B of Lemma 4 below). However, we first describe how we obtain the 4-ranks of our quadratic number fields.      

     To obtain the 4-ranks of the class groups of our various fields, we will utilize both

d_k-splittings of the second kind and Rédei matrices (cf. [15], [16], ]18], [19], [20]). Given an algebraic number field k, we define a d_k-splitting of the second kind to be a splitting of d_k­ into two relatively prime fundamental discriminants (d₁, d₂) such that the Kronecker symbols

(d₁/p) = 1 and (d₂/q) = 1 for all primes p dividing d₂ and for all primes q dividing d₁, and where we consider the splitting (1, d₁d₂) to be the trivial d_k-splitting of the second kind. It is

well known that for any quadratic (number field k, if s is the number of d_k-splittings of the second kind and r is the 4-rank of the narrow-class group of k, then 2^r = s (cf. [16]).

     We also use Rédei matrices, which are described as follows (cf. [16], [17], [19], [20], [21]), where k is a quadratic number field. Let d_k = p₁*p₂*…p_t* be the unique factorization of d_k into a product of prime discriminants, where p* = (-1)^(p-1)/2)p if p is an odd prime, and p* = -4, 8, or -8 if p = 2. The Rédei matrix R_k = (a_ij) with entries in the finite field F₂, is describedas a matrix consisting of Kronecker symbol representations a_ij such that -1 to the exponent a_ij equals   {(p_i*/p_j) if i ≠ j, and ((d_k/p_i*)/p_i) if i = j}. For properties of Rédei matrices see the above references, and in particular we mention that the sum of all row vectors of R_k is equal to the   zero vector in F₂, and the following 4-rank property of Rédei matrices that we will make much use of to establish our results in this paper.

 

Rédei matrix 4-rank property: The 4-rank of the narrow class group of a quadratic         number field k is equal to t – 1 – rank R_k, where t is the number of prime discriminants     dividing d_k, and if k is imaginary this translates into the 4-rank of C_k is equal to t – 1 – rank R_k.  

 

     We now state our four lemmas mentioned above.

Lemma 4: Let k be an imaginary quadratic number field such that rank C_k,2 = 4. Then k has infinite 2-class field tower in the following cases:                

A) there exists a negative prime discriminant d_j dividing d­_k, such that (-p_j/p_i) = 1 where p_j is the prime dividing d_j, all p_i are distinct, and p_i is distinct from p_j for all primes p_i dividing d_k,

1 ≤ i ≤ 5;

B) for some prime p_j congruent to 1 mod 4, or p_j = 2, in which case we further assume that 8 is a fundamental discriminant dividing d_k, we have (p_j/p_k) = (p_k/p_l) = (p_j/p_m) = 1, p_j, p_k, p_l, p_m distinct

primes dividing d_k;

C) at least two of the prime discriminants dividing d_k are positive, and (p₁/p₃) = (p₂/p₃) = 1 where p₁ and p₂ are distinct primes dividing positive prime discriminants dividing d_k, and p₃ is a prime dividing a positive or negative prime discriminant dividing d_k, where p₃ is not equal to p₁ or p₂.  

Remark 1: We note that in Lemma 1 of [2] we mistakenly stated the requirement in Case B above as (p_j/p_k) = (p_k/p_l) = (p_j/p_m) (see Lemma 1 in [1] for the correct statement).

Lemma 5: Let k be an imaginary quadratic number field with rank C_k,2 = 4 and 4-rank of C_k

equal to 2. Then k has infinite 2-class field tower in each of the following cases:

A) five negative prime discriminants divide d_k;

B) d_k not congruent to 4 mod 8;

C) d_k congruent to 4 mod 8 and exactly three negative prime discriminants divide d_k,

and the Kronecker symbols of the primes dividing d_k do not have the format (p₁/q₁) = (p₁/q₂) = (p₁/q₃) = (p₂/q₁) = (p₂/q₂) = (p₂/q₃) = -1, where p₁ and p₂ are distinct primes dividing positive prime discriminants dividing d_k, and q₁, q₂, and q₃ are distinct primes dividing negative prime discriminants dividing d_k.  

   For our next lemma, we note that If k is an imaginary quadratic number field with 4-rank 1, then from our above description of d_k-splittings of the second kind we see that there is a unique non-trivial d_k-splitting of the second kind for k. The following lemma is a corrected version of Lemma 6 in [2] from our earlier work (see Remark 2 below), in which we use the following notation: p₁^* and p₂^* are distinct positive prime discriminants dividing d_k, q₁^* is a negative prime discriminant dividing d_k, and r_i^*, 2 ≤ i ≤ 5, represent distinct positive prime discriminants or negative prime discriminants dividing d_k such that r_i^* is distinct from p₁^*, p₂^*, and q₁^*.  

Lemma 6: Let k be an imaginary quadratic number field with rank C_k,2 = 4 such that C_k has     4-rank 1. If the nontrivial d_k-splitting of the second kind is either (without loss of generality)   (q­­₁^*, r₂^*r₃^*r₄^*r₅^*) or (p₁^*, r₂^*r₃^*r₄^*r₅^*), or if d_k is not congruent to 4 mod 8 and the nontrivial         d_k-splitting of the second kind is (p₁^*p₂^*, r₁^*r₂^*r₃^*), then k has infinite 2-class field tower.

Remark 2: We note that in the statement of Lemma 6 in [2] we mistakenly included the case
d_k congruent to 4 mod 8 with the nontrivial d_k-splitting of the second kind (p₁^*p₂^*, r₁^*r₂^*r₃^*).

For parts i and ii of the following lemma, cf. [19], [20]: Proposition: parts ii and iv.  

Lemma 7: Let k be an imaginary quadratic number field and let F be subfield of the genus field of k.

i) Suppose that F/Q is a totally real biquadratic extension, where Q is the field of rational numbers. If three rational primes split completely in F and another rational prime is unramified in F and these four rational primes ramify in k, then the 2-class field tower of k is infinite.

ii) Suppose that F/Q is a totally imaginary biquadratic extension. If two rational primes split completely in F and another rational prime is unramified in F and these three ramified primes

ramify in k, then the 2-class field tower of k is infinite.

iii) Let k be an imaginary quadratic number field such that rank C_k,2 = 4, C_k has 4-rank 1,

five negative prime discriminants divide d_k, and d_k is not congruent to 4 mod 8. Then k has infinite 2-class field tower, where Q is the field of rational numbers.

Remark 3: Recall the well-known result from genus theory (cf. [3], [8], [16]) that if k is a quadratic number field with discriminant d_k and t is the number of primes that ramify in k (which is the number of primes that divide d_k), then rank C_k,2 = t – 2 if d_k > 0 and is not a sum of two squares, and rank C_k,2 = t – 1 otherwise.

 

          We also state a result from Mouhib (2010), that he utilized to demonstrate that there are

imaginary quadratic number fields k with rank C_k,2 = 2 and 4-rank of C_k equal to 2 such that      

k has infinite 2-class field tower (see Remark 11 in the final section). We discuss this result in regard to an open question about the 2-class field tower conjecture (see Question 4 in the     Open Questions section).

Lemma 8: Let d be a square-free positive integer such that d is not congruent to 1 mod 4,

let M = Q(√d), and assume that 8 divides C_M. Then for every prime number q congruent to

3 mod 4 such that the equation x² – dy² = q has a solution in Z x Z (where Z is the ring of

integers), the imaginary quadratic number field Q(√-qd) has infinite 2-class field tower.

 

Remark 4: We note that given the assumptions of Lemma 8, it must be the case that

x and y are both odd, and d is congruent to 2 mod 4.

 

 

An Application of a Result by Schmithals

to Obtain New Imaginary Quadratic Number Fields k with Rank C_k,2 = 4

that Satisfy the 2-Class Field Tower Conjecture

     To find new fields k that satisfy the 2-class field tower conjecture we now utilize the following application of a result by Schmithals (cf. the inequalities (1) and (3) in [17]) where k is an imaginary quadratic number field, F is a real quadratic number field, K is the compositum of

k¹ and F, h denotes the 2-class number (in the wide sense) of F, and m is the number of primes q dividing d_k such that (d_F/q) = -1.

 

Lemma 9: If m ≥ (1/h)(3 + 2√(2h + 1) then k has infinite 2-class field tower.

 

Remark 5: Lemma 9 follows immediately from Schmithals’ more generic inequalities in [17] (cf. inequalities (1) and (3)). As noted by Schmithals, a more generic form of Lemma 9

(cf. inequality (3) in [17]) is satisfied with m = 1 and h ≥ 16.

 

 

Remark 6: Schmithals utilized what we have referred to as Lemma 9 to prove that there are infinitely many imaginary quadratic number fields k with exactly three primes dividing d_k

that have infinite 2-class field tower (cf. Satz 1 and Beispiel 4, which is k = Q(√-5.11.461),

in [17]). Utilizing our above Remark 3 and 4-rank of C_k criteria, we see that in effect Schmithals proved in 1980 that there are infinitely many imaginary quadratic number fields k with

rank C_k,2 = 2 and 4-rank of C_k equal to 2 that have infinite 2-class field tower. Mouhib in 2010 proved this same result (cf. Prop. 3.3 in [14]), referring to Schmithals’ example

k = Q(√5.11.461), but he did not mention Schmithals’ essential proof of the result. For historical accuracy we would like to emphasize here that the proof of this result should be attributed to Schmithals in 1980.

 

     Utilizing Lemma 9 we immediately obtain the following lemma, which is what we utilize to obtain our new imaginary quadratic number fields k with rank C_k,2 = 4 and infinite 2-class field tower.     

 

Lemma 10: Let p_i, q_i, i = 1, 2, 3, 4, be distinct prime numbers such that p_i is congruent to           1 mod 4 and q_i is congruent to 3 mod 4, and let q₅ be a prime number such that q₅ is congruent to
3 mod 4, or q₅ = 2 if q₅* = -4 or q₅* = -8, and q₅ is not equal to q_i for i = 1, 2, 3 or 4. Without loss of generality let M = Q(√p₁p₂q₁) (resp. Q√q₁q₂q₃), Q√q₁q₂q₃q₄), Q(√p₁p₂q₁q₂), Q√2q₁q₂q₃), Q(√2p₁q₁q₂), Q(√2p₁p₂q₁), Q(√2p₁p₂p­₃), Q(√p₁p₂p­₃p₄)). Assume that 16 divides h(M), and (4p₁p₂q_1­­­/q₅) = -1 (resp. (4q₁q₂q₃/q₅) = -1, (q₁q₂q₃q₄/q₅) = -1, (p₁p₂q₁q₂/q₅) = -1, (2q₁q₂q₃/q₅) = -1, (2p₁q₁q₂/q₅)= -1, (2p₁p₂q₁/q₅) = -1, (2p₁p₂p­₃/q₅) = -1, (p₁p₂p­₃p₄/q₅) = -1). Let L be an imaginary quadratic number field with exactly five primes dividing d_L, and moreover let L = Q(√-p₁p₂q₁q₅) (resp. Q√-q₁q₂q₃q₅), Q√-q₁q₂q₃q₄q₅), Q√-q₁q₂q₃q₄), Q(√-p₁p₂q₁q₂q₅), Q(√-p₁p₂q₁q₂),               Q√-2q₁q₂q₃q₅), Q(√-2p₁q₁q₂q₅), Q(√-2p₁p₂q₁q₅), Q(√-2p₁p₂p­₃q₅), Q(√-2p₁p₂p­₃p₄),                   Q(√-p₁p₂p­₃p₄q₅), Q(√-p₁p₂p­₃p₄)). Then L has infinite 2-class field tower.

 

     In the next section we will demonstrate by the use of Lemma 10 the existence of some new imaginary quadratic number fields k such that rank C_k,2 = 4 and C_k has 4-rank 2, which satisfy the 2-class field tower conjecture as described above. For the new fields that we will obtain,  the Kronecker symbols listed in Case C of Lemma 5 all have value -1, and these fields satisfy the discriminant conditions of Lemma 5; in particular d_k is congruent to 4 mod 8, exactly three negative prime discriminants divide d_k, and the Kronecker symbols of the primes dividing d_k have the format (p₁/q₁) = (p₁/q₂) = (p₁/q₃) = (p₂/q₁) = (p₂/q₂) = (p₂/q₃) = -1, where p₁, p₂, q₁, q₂, and q₃ are defined as in Case C of Lemma 5. To obtain these new fields, we make use of    Rédei matrices as we have described above (cf. [18], [19], [20]), and we give a particular formulation in the context of the following lemma.

 

Lemma 11: Let k be an imaginary quadratic number field such that rank C_k,2 = 4 and C_k has

4-rank 2, exactly three negative prime discriminants divide d_k, d_k is congruent to 4 mod 8, and the Kronecker symbols of the primes dividing d_k satisfy the equalities listed in Case C of

Lemma 5. Without loss of generality let q₁ = 2, and M₁, M₂, and M₃ denote the corresponding real quadratic fields M₁ = Q(√p₁p₂q₂q₃), M₂ = Q(√p₁p₂q₂), and M₃ = Q(√p₁p₂q₃). Then

h(M₂) = h(M₃) = 4, and h(M₁) ≥ 8; h(M₁) ≥ 16 if and only if the narrow class group of M₁ has

8-rank 1. If (p₁/p₂) = 1 and the narrow class group of M has 8-rank 1, then k is a new field with infinite 2-class field tower.

Proof: We see from Sueyoshi (cf. [19], [20]) that the Rédei matrices of fields k that satisfy the assumptions of the lemma can be described as follows, where the entries a_ij are in the field F₂,   1 ≤ i ≤ 5, 1 ≤ j ≤ 5; if i ≠ j then a_ij = 1 if and only if (r_i/r_j) = -1, and if i = j then a_ij = 1 if and only if [d_k/r­_i^*]/r_i) = -1, where r_i,1 ≤ i ≤ 5, are the primes dividing d_k, r₁^* = -4, r₁ = q₁ = 2, r₂^* < 0,

r₃^*< 0, r₄^* > 0, r₅^* > 0, and [d_k/r_i^*] denotes d­­_k divided by the prime discriminant r_i^*; the symbol * is used to denote that a_ij can have the value 0 or 1, with the stipulations described below;         we denote r₂ = q₂, r₃ = q₃, r₄ = p₁, and r₅ = p₂.

Then R_k = [*     1     1     0   0                                                                                                                          

                 *     1     1     1     1

                 *     0     0     1     1

                 1     1     1     *   *

                   1     1     1     *     *]

     As described by Sueyoshi (cf. [19]), it is understood that this Rédei matrix formulation of R_k can have a change of order of the r_i’s, 2 ≤ i ≤ 5, the sum of all row vectors of R_k is equal to the zero vector in F₂, and if a₄₄ = 0 then a₄₅ = a₅₄ = a₅₅ = 0 and a₁₁ = 1, and if a₄₄ = 1 then a₄₅ = a₅₄ =     a₅₅ = 1 and a₂₁ = 1; consequently there are four generic types of these Rédei matrices. From an examination of the matrices of M₁, M₂, and M₃, we see that the ranks of these matrices are respectively 2, 3, and 3, and from Remark 3 and the 4-rank formula for the narrow class group using Rédei matrices (see Preliminaries), along with the well-known fact that if a prime congruent to 3 mod 4 divides d_k then the rank of the narrow group is twice the rank of the wide class group (see for example [16]), we obtain that the 2-class groups of M₂ and M₃ are elementary (i.e. have 4-rank 0), and the 2-class group of M₁ has 4-rank 1, which implies that h(M₂) = h(M₃) = 4 and h(M₁) ≥ 8. It is immediate from what we have described above that the condition h(M₁) ≥ 16 is equivalent to the narrow class group of M₁ having 8-rank 1, which is equivalent to the (wide) class group of M₁ also having 8-rank 1, and there is a known criteria using the Hilbert symbol to determine the 8-rank of a quadratic number field (cf. [21]). Finally, if (p₁/p₂) = 1 (meaning that (r₄/r₅) = 1) we see from the description of R_k given above that a₁₁ = 1 and consequently we must have a₂₁ ≠ a₃₁. Therefore we obtain the Kronecker symbol               (p­­₁p₂q₂q₃/2) = -1, which along with the assumption that the narrow class group of M₁ has           8-rank 1 (which as we have seen, is equivalent to h(M₁) ≥ 16) gives us the required conditions of Lemma 10 to obtain that k has infinite 2-class field tower. Since we are not able to utilize either Lemma 4 or Lemma 7 (or any other formulations in the literature to the best of our knowledge; see Case 60 in [20]) to show that k has infinite 2-class field tower, k is a “new” field and our lemma is established.

 

NOTE: For the remainder of this paper, unless stated otherwise k will always denote an imaginary quadratic number field with rank C_k,2 = 4.                    

     The following lemma is useful to show that if a field L satisfies the conditions of Lemma 11, then there are infinitely many such fields that also satisfy these conditions, and consequently have infinite 2-class field tower, which we will make use of to show that there are infinitely many new fields k that satisfy the 2-class field tower conjecture when C_k has 4-rank 1.

Lemma 12: Assume there exists a field L that satisfies the conditions of Lemma 10. Then there exist infinitely many such fields that also satisfy these conditions, and therefore there exist infinitely many such fields that have infinite 2-class field tower that satisfy the conditions of Lemma 10.

Proof: By way of illustration, we demonstrate the result for two specific cases of Lemma 10, using the Chinese Remainder Theorem (CRT) and Dirichlet’s Theorem of Primes in an Arithmetic Progression (DPAP). Let M = Q(√p₁p₂q₁) and L = Q(√-p₁p₂q₁q₅). Using CRT and DPAP we are able to formulate infinitely many fields L of the form Q(√-p₁p₂q₁q) where q is a prime congruent to 3 mod 4 and (p₁/q₅) = (p₁/q), (p₂/q₅) = (p₂/q), and (q₁/q₅) = (q₁/q).            Now let M = Q(√p₁p₂q₁q₂) and L = Q(√-p₁p₂q₁q₂). Similarly, we once again use CRT and DPAP, to formulate infinitely many fields L of the form Q(√-p₁p₂q₁q₂q) where q is a prime congruent to 3 mod 4, and (p₁/2) = (p₁/q), (p₂/2) = (p₂/q), (-q₁/2) = (2/q₁) = (q₁/q), and                (-q₂/q) = (2/q₂) = (q₂/q). Since (p₁p₂q₁q₂/2) = (p₁p₂q₁q₂/q), we see that we again have obtained infinitely many fields that satisfy the conditions of Lemma 10, and consequently have infinite    2-class field tower. The remaining cases of Lemma 10 are done similarly, and we leave the details to the reader.                          

 

Remark 7: Our use of CRT and DPAP in the proof of Lemma 12 generalizes the technique Mouhib used in the rank C_k,2 = 2 case, as described in the remark following Prop. 3.3 in [14] (however, see Remark 6 above). Note that if k is a field described in Lemma 11 for which we are able to show has infinite 2-class fields tower by means of Lemma 10, then we must have M₁ = Q(√p₁p₂q₂q₃). Therefore when we use CRT and DPAP as described in the proof of Lemma 12, the infinitely many fields we obtain all have d_k not congruent to 4 mod 8, and therefore by Lemma 5: Part b we know that any such field k in this infinite collection for which C_k has

4-rank 2 is not a new field.  

     We now demonstrate the existence of new fields k with infinite 2-class field tower such that

rank C_k,2 = 4 and C_k has 4-rank 2, three negative prime discriminants divide d_k, d_k is congruent to      

4 mod 8, and the Kronecker symbols of the primes dividing d_k satisfy the equalities listed in

Case C of Lemma 5.

 

Case 1: 4-Rank of C_k Equal to 2

    We see from above that in the 4-rank of C_k equal to 2 case we know that k has infinite         2-class field tower except for one family of fields, and from [2] we see that this is specifically the family of fields such that d­_k is congruent to 4 mod 8, exactly three negative prime discriminants divide d_k, and the Kronecker symbols of the primes dividing d_k have the format (p₁/q₁) = (p₁/q₂) = (p₂/q₂) = (p₂/q₂) = (p₁/2) = (p₂/2) = -1, where p₁ and p₂ are primes congruent to 1 mod 4 dividing d_k, and q₁ and q₂ are primes congruent to 3 mod 4 dividing d_k. To make use of    Lemma 11 to obtain our new fields, we begin by defining a Rédei sub-type matrix of 4-rank n,  n = 0, 1, or 2, which we denote by R_n, to be a matrix with particular conditions that is contained in a family S of generic Rédei matrices listed in [19] or [20], such that if k is a field with the property that the primes dividing d_k satisfy the Kronecker symbol matrix entries of R_n        (which we will denote by saying that “k satisfies R_n”), then C_k has 4-rank n and k does not satisfy any matrix in S – R_n. Once again the matrix entry * signifies a choice of 0 or 1, and when d_k is congruent to 4 mod 8 it is understood that q₁* = -4. It is also understood that these        Rédei matrices are generic in the sense that aside from q₁* = -4, the order of the primes dividing positive prime discriminants (resp. dividing negative prime discriminants) can be changed       (cf. [18]. [19], [20]). We refer to a particular Rédei sub-type matrix R_n as “open” if there exists

a field k as above that satisfies R_n for which it is not known if k is a new field with infinite          

2-class field tower.

     From Case 60 in [20] we see that there are exactly four open Rédei sub-type matrices of         4-rank 2, and that if k satisfies any one of these four matrices then k has the above Kronecker symbol format. We refer to fields k with 4-rank 2 that have the above Kronecker symbol format as Family D₂ fields. From [20] we see that a₂₃ = a₄₁ =a₅₁ = a₂₄ = a₂₅ = a₃₄ =a₃₅ =1 for all four of these open Rédei sub-type matrices, and that we can distinguish these matrices in the following way (see the above description of Rédei matrices):

a) a₄₅ = 0, a₁₁ = 1, a₂₁ = 1, a₃₁ = 0                                                                                                                       b) a₄₅ = 0, a₁₁ = 1, a₂₁ = 0, a₃₁ = 1                                                                                                           c) a₄₅ = 1, a₁₁ = 1, a₂₁ = 1, a₃₁ = 0                                                                                                          d) a₄₅ = 1, a₁₁ = 0, a₂₁ = 1, a₃₁ = 1                                                                                                      

We can describe the generic Rédei matrix R_k for the D₂ family as follows, where the * entries would be entered in accordance with the above stipulations for matrices a, b, c, and d (cf. [20]):

[*     1     1     0     0

*     1     1     1     1

*     0     0     1     1

1     1     1     *     *

1     1     1     *     *]

 

     In our examples that followwe utilize Lemmas 10 and 11 to demonstrate the existence of some new fields k as above that have infinite 2-class field tower and satisfy cases a, b, or c above, and we use [4] to obtain the 2-class numbers of our corresponding real quadratic    number fields M.

Example 1: k = Q(√-5.13.7.827) = Q(√-376285), M = Q(√5.13.7.827) = Q(√376285), h(M) = 16, (5.13.7.827/2) = -1, (5/2) = (13/2) = (5/7) = (5/827) = (13/7) = (13/827) = -1, C_k has 4-rank 2 and the three nontrivial d_k-splittings of the second kind of k are (5.13, -4.7.827), (-5.827, 4.7.13), and (-13.827, 4.5.7). Since from Lemma 10 we see that k is a new field with infinite 2-class field tower and this cannot be obtained by using either Lemma 4 or Lemma 7, we can conclude that   k is a new field of this type (see Case 60 in [20]).

Example 2: k = Q√-5.13.47.827) = Q(√-2526485), M = Q√5.13.47.827) = Q(√2526485),     h(M) = 16, (5.13.47.827/2) = -1, (5/2) = (13/2) = (5/47) = (5/827) = (13/47) = (13/827) = -1,   C_k has 4-rank 2. Since (2/7) = (2/47) = (827/7) = (827/47) = 1, we see from the above Kronecker symbol equalities that the primes 7 and 47 have the same Kronecker symbol formulations with the other primes dividing d_k, and from Lemma 10 we obtain a new field with infinite 2-class field tower.

Example 3: k = Q(√(-5.29.47.827) = Q(√-5636005), M = Q(√5.29.47.827) = Q(√5636005), h(M) = 16; C_k has 4-rank 2. Since (29/5) = 1 we see immediately from the proof of Lemma 11 that (5.29.47.827/2) = -1, and from Lemma 10 or Lemma 11 we are able to conclude that k is a     new field with infinite 2-class field tower.

Example 4: Let k = Q(√-5.29.47.43) = Q(√-293045), and M = Q(√5.29.47.43) = Q(√293045).     We see that h(M) = 16, (5.29.47.43/2) = -1, and in a similar manner to Examples 1, 2, and 3    we are able to conclude from Lemma 10 that k has infinite 2-class field tower.

Remark 8: Although our above examples illustrate that we can find new fields that satisfy the Rédei matrices listed in cases a, b, and c above, we are not able to utilize Lemma 10 to find a  new field with infinite 2-class field tower that satisfies the Rédei matrix listed in case d.        This can be seen from the fact that (p₁p₂q₂q₃/2) = 1, and from the fact that h(Q(√p₁p₂q₂)) = h(Q(√p₁p₂q₃)) = 4 (cf. Lemma 10).                                                                                                                                                                                                                                                                                                                                       

   We now put together Lemmas 10 and 11 and our above examples to state our first main result as the following theorem.

 

Theorem 1: Let F_k be the family of imaginary quadratic number fields k such that rank C_k,2 = 4, C_k has 4-rank 2, d_k is congruent to 4 mod 8, three negative prime discriminants divide d_k, and the Kronecker symbols of the primes dividing d_k have the format (p₁/q₁) = (p₁/q₂) = (p₁/q₃) = (p₂/q₁) = (p₂/q₂) = (p₂/q₃) = -1, where p₁ and p₂ are distinct primes dividing positive prime discriminants dividing d_k, and q₁, q₂, and q₃ are distinct primes dividing negative prime discriminants dividing d_k.Then there exist new fields k in this family that have infinite 2-class field tower.

 

     For the case when rank C_k,2 = 4 and 4-rank of C_k equal to 1, we know from Sueyoshi

(cf. Lemma 7: Part iii and [19]) that if five negative prime discriminants divide d_k and d_k is not congruent to 4 mod 8 then k has infinite 2-class field tower. For each of the remaining cases, i.e. when d_k is congruent to 4 mod 8 and exactly three or five negative prime discriminants divide d_k, and when exactly three discriminants divide d_k and d_k is not congruent to 4 mod 8,

we will show that there exists at least one new imaginary quadratic number field k with

infinite 2-class field tower. Furthermore, we will show that there are infinitely many new such fields k with infinite 2-class field tower in the case when exactly three discriminants divide d_k and d_k is not congruent to 4 mod 8. We begin with the case when five negative prime discriminants divide d_k and d_k is congruent to 4 mod 8.

 

Case 2: Five Negative Prime Discriminants Dividing d_k

with 4-Rank of C_k Equal to 1 and d_k Congruent to 4 mod 8

     For the case when five negative prime discriminants divide d_k, the 4-rank of C_k is equal to 1, and d_k is congruent to 4 mod 8, we see from [19] that there are exactly six particular open Rédei sub-type matrices of 4-rank 1, and that k always has infinite 2-class field tower for two of these          Rédei sub-type matrices (corresponding to matrices m and o in [19]). For the remaining two generic Rédei sub-type matrices (corresponding to matrices n and p in [19]) we now make use of Lemma 11 to show that there exists a new field k with infinite 2-class field tower that satisfies matrix n (resp. matrix p). We describe matrices n and p as follows, where once again it is understood that p₁* = -4, the matrix entry * signifies a choice of 0 or 1, and that these          Rédei matrices are generic in the sense that the order of the q­_i’s (2 ≤ i ≤ 5) can be changed.    We divide each of the matrices n and p into its two possible generic Rédei matrices: n_1, n₂, p₁, p₂ as follows:

 

 

n₁: [1   1   1   1   1       n₂: [0   1   1   1   1       p₁: [*   1   1   1   1       p₂: [*   1   1   1   1          

       1  1   1   1   1             0   1   1   1   1             *   0   1   1   0             *   0   1   1   0

       *   0   1   1   0             *   0   1   1   0             1   0   0   1   1             0   0   0   1   1

       *   0   0   1   1            *   0   0   1   1             1   0   0   1   1              0   0   0   1   1

       *   0   1   0   1]           *   0   1   0   1]             *   1   0   0   1]            *   1   0   0   1]

     From the table on pages 337-338 in [19] we see that for matrix p₂ it is known that k has infinite 2-class field tower, and therefore we eliminate matrix p₂ in our subsequent formulations; furthermore we see that matrices p_1, n₁, and n₂ are open Rédei matrices.

   In order to make use of Lemma 10 to find new fields k that satisfy Rédei matrices n₁, n₂, and p₁, our next lemma demonstrates that we must have d_M congruent to 4 mod 8 for the corresponding real quadratic number field M given in Lemma 10.

Lemma 13: Let k satisfy Rédei matrices n₁, n₂, or p₁, and let M be any of the corresponding      real quadratic number fields given in Lemma 10. If d_M is not congruent to 4 mod 8 then C_M,2 is isomorphic to the group Z/2Z x Z/2Z.  

Proof: If d_M is not congruent to 4 mod 8, then d_M = q₂.q₃.q₄.q₅ for distinct primes q_i congruent to 3 mod 4, 2 ≤ i ≤ 5. For Rédei matrices n₁ and n₂ we have the following Rédei matrix:

R_M = [0   1   1   1        

           0   0   1   0

           0   0   0   1

           0   1   0   0]

     Since R_M has rank 3 we see from the above Rédei matrix 4-rank property that C_M has            4-rank 0, and therefore by genus theory C_M,2 is isomorphic to Z/2Z x Z/2Z.                              For Rédei matrix p₁ we have the following Rédei matrix:

R_M = [1   1   1   1        

           0   1   1   1

           0   0   0   1

           1   0   0   0]

     Once again we see that R_M has rank 3 and consequently C_M,2 is isomorphic to Z/2Z x Z/2Z, which establishes our lemma.

     We now show that when d_M is congruent to 4 mod 8 there exists a new field k that satisfies Rédei matrix n­₁ (resp. n₂, p₁).

Theorem 2: There exist new imaginary quadratic number fields k such that rank C_k,2 = 4,        C_k has 4-rank 1, five negative prime discriminants divide d_k, and d_k is congruent to 4 mod 8.    In particular, for each possible open Rédei sub-type matrix of this family: n₁, n₂, p₁, there exists a new field k that satisfies the given open Rédei sub-type matrix.

Proof: We show that for each of n₁, n₂, p₁, there exists a new field k that satisfies the conditions of the theorem and that satisfies the given open Rédei sub-type matrix, which will prove our result. To establish our result we utilize Lemma 10, which necessitates finding a corresponding real quadratic number fields M to k such that h(M) ≥ 16 and for which the Kronecker symbol condition given in Lemma 10 is satisfied. From Lemma 13 we know that d_M must be congruent to 4 mod 8. The following fields respectively satisfy each of n₁, n₂, and p₁, and from [4] we see that they also satisfy the 2-class number bound condition of Lemma 10, with h(M) = 16, where          k = Q(√-q₂q₃q₄q₅) and (without loss of generality) M = Q(√q₂q₃q₄).

     for p₁: M = Q(√23.19.67) = Q(√29279)

     for n₁: M = Q(√11.7.167) = Q(√12859)

     for n₂: M = Q(√23.11.19) = Q(√4807)

     By choosing our remaining prime q₅ appropriately, which we do in Examples 5, 6, and 7 below, we are also able to obtain the second condition of Lemma 10, namely that              (q₂q₃q₄/q₅) = -1, and therefore we are able to formulate new fields k = Q(√-q₂q₃q₄q₅) that respectively satisfy each of n₁, n₂, and p₁, and this proves our theorem.      

     As described in the proof of Theorem 2, for each of the following fields k = Q(√-q₂q₃q₄q₅) we have h(M) = 16 and (q₂q₃q₄/q₅) = -1, where M = Q(√q₂q₃q₄), and consequently the conditions of Lemma 10 are satisfied and we can conclude that k is a new field with infinite 2-class field tower.

Example 5: for p₁: k = Q(√-23.19.67.3) = Q(√-87837); q₅ = 3, (-23/3) = 1, (-19/3) = (-67/3) = -1, (23.19.67/3) = -1, h(Q(√23.19.67)) = 16

Example 6: for n₁: k = Q(√-11.7.167.79) = Q(√-1015861); q₅ = 79, (-17/79) = (-167/79) = -1,   (-7/79) = 1, (11.7.167/79) = -1, h(Q(√11.7.167)) = 16

Example 7: for n₂: k = Q(√-23.11.19.103) = Q(√-495121); q₅ = 103, (-23/103) = (-19/103) = -1, (-11/103) = 1, (23.11.19/103) = -1, h(Q(√23.11.19)) = 16

 

Case 3: Exactly Three Negative Prime Discriminants Dividing d_k

with 4-Rank of C_k Equal to 1 and d_k Congruent to 4 mod 8

     For the case when exactly three negative prime discriminants divide d_k, the 4-rank of C_k        is equal to 1, and d_k is congruent to 4 mod 8, we see from [20] that there are 34 open            Rédei sub-type matrices of 4-rank 1, which are listed as belonging to Cases 56, 57, 58, 59, and 60 in [20]. We also see from [20] that all fields k with the Kronecker symbol format we described above in the 4-rank 2 case as designating Family D₂ fields, i.e. (p₁/q₂) = (p₁/q₃) = (p₂/q₂) = (p₂/q₃) = (p₁/2) = (p₂/2) = -1 where p₁ and p₂ are primes congruent to 1 mod 4 dividing d_k, and q₂ and q₃ are primes congruent to 3 mod 4 dividing d_k, satisfy a Rédei sub-type matrix in Case 60 of [20].  We now designate fields of this type for the 4-rank 1 case as Family D₁ fields, and fields for the 4-rank 1 case that satisfy an open Rédei sub-type matrix in Case 60 as Family D fields. We begin by establishing the following lemma, where the primes p_1, p_2, q_2, q₃ are as above, and once again the negative prime discriminant q₁* = -4.

Lemma 14: Assume that exactly three negative prime discriminants divide d_k, the 4-rank of C_k is equal to 1, d_k is congruent to 4 mod 8, and that k satisfies an open Rédei sub-type matrix. Let M_1, M₂, and M₃ denote the three corresponding real quadratic number fields given in Lemma 10 as follows (without loss of generality): M₁ = Q(√q₂p₁p₂), M₂ = Q(√q₃p₁p₂), M₃ = Q(√p₁p₂q₂q₃). Then h(M₁) = h(M₂) = 4; and h(M₃) ≥ 8 if and only if k is a Family D field.

Proof: We give an illustration of the method by initially describing the five open Rédei           sub-type matrices of 4-rank 1 belonging to Case 57 in [20] as follows, where a₅₄ = 1, a₂₁ = a_51:          a) a₄₁ = 0, a₅₁ = 1, a₃₁ = 1                                                                                                                     b) a₄₁ = 0, a₅₁ = 1, a₃₁ = 0                                                                                                                        c) a₄₁ = 1, a₅₁ = 0, a₃₁ = 1                                                                                                                        d) a₄₁ = 1, a₅₁ = 0, a₃₁ = 1                                                                                                                         e) a₄₁ = 1, a₅₁ = 1, a₃₁ = 1  

We describe the generic Rédei matrix R_k for this family below, where the * entries would be entered in accordance with the above stipulations for matrices a, b, c, d, and e (cf. [20]):

[*     1     1     0     0

*     0     1     1     0

*     0     1     0     1

*     1     0     0     1

*     0     1     1     0]

It follows that the Rédei matrices of M_1, M₂, and M₃ for each of the five members of this family all have rank 3, and therefore in accordance with genus theory and the Rédei matrix 4-property we are able conclude that the 2-class groups of M_1, M₂, and M₃ are isomorphic to Z/2Z x Z/2Z for each member of this family. We illustrate this result by displaying M_1, M₂, and M₃ for the       Rédei matrix of Case 57a, and we leave it to the reader to verify this for the remaining cases.

 

M₁ = [0     1     0     0               M₂ = [0     1     0     0           M₃ = [1     1     1     0          

           1     0     1     0                         1     0     0     0                     0     0     0     1

           0     1     0     1                          0     0     1     1                     1     0     0     1

           1     0     1     1]                         1     1     1     0]                     0     1     1     1]

In a similar manner it can be readily shown that the same result applies to Cases 56, 58, and 59 as we have indicated applies to Case 57. However, for Case 60 with k as in the lemma,           we demonstrate that we always obtain h(M₁) = h(M₂) = 4, h(M₃) ≥ 8.

   We begin by designating the ten open Rédei sub-type matrices of 4-rank 1 belonging to     Case 60 in [20] as follows, where a₂₃ = a₂₄ = a₂₅ =a₃₄ = a₃₅ = 1:

a) a₄₅ = 0, a₄₁ = a₅₁ = 1, a₁₁ = 0, a₂₁ = a₃₁ = 1                                                                                  

b) a₄₅ = 0, a₄₁ = a₅₁ = 1, a₁₁ = 0, a₂₁ = 0, a₃₁ = 0                                                                                  

c) a₄₅ = 0, a₄₁ = 1, a₅₁ = 0, a₁₁ = 0, a₂₁ = 1, a₃₁ = 0                                                                                 d) a₄₅ = 0, a₄₁ = 1, a₅₁ = 0, a₁₁ = 0, a₂₁ = 0, a₃₁ = 1                                                                                 e) a₄₅ = 0, a₄₁ = 1, a₅₁ = 0, a₁₁ = a₂₁ = a₃₁ = 1                                                                                          f) a₄₅ = 0, a₄₁ = 1, a₅₁ = 0, a₁₁ = 1, a₂₁ = a₃₁ = 0                                                                                     g) a₄₅ = a₄₁ = a₅₁ = 1, a₁₁ = 1, a₂₁ = 0, a₃₁ = 1                                                                                        h) a₄₅ = a₄₁ = a₅₁ = 1, a₁₁ = a₂₁ = a₃₁ = 0                                                                                                 i) a₄₅ = 1, a₄₁ = 0, a₅₁ = 1, a₁₁ = 1, a₂₁ = a₃₁ = 0                                                                                        j) a₄₅ = 1, a₄₁ = 0, a₅₁ = 1, a₁₁ = a₂₁ = 0, a₃₁ = 1

We describe the generic Rédei matrix R_k for this family below, where the * entries would be entered in accordance with the above stipulations for matrices a through j (cf. [20]):

[*     1     1     0     0

*     0     1     1     1

*     0     1     1     1

*     1     1     *     *

*     1     1     *     *]

It readily follows that for all the above fields we obtain h(M₁) = h(M₂) = 4, h(M₃) ≥ 8, and we illustrate this result by displaying M_1, M₂, and M₃ for the Rédei matrix of Case 60g.

M₁ = [0     1     0     0               M₂ = [1     1     0     0           M₃ = [0     1     1     1          

           0     1     1     1                          1     1     1     1                     0     1     1     1

           1     1     0     1                         1     1     0     1                     1     1     1     1

           1     1     1     0]                         1     1    1     0]                     1     1     1     1]

Since M₁ and M₂ have ranks 3 and M₃ has rank 2, we see from genus theory and the                   Rédei matrix 4-property that the 2-class groups of M₁ andM₂ are isomorphic to Z/2Z x Z/2Z, and that M₃ has 4-rank 1 and consequently has order greater than or equal to 8. In a similar way      it can readily be shown that this same result applies to all members of the above family          (we once again leave the remaining cases for the reader to verify), which establishes our lemma.

     We now demonstrate the existence of a new field in the case when exactly three negative prime discriminants divide d_k, the 4-rank of C_k is equal to 1, and d_k is congruent to 4 mod 8.

Theorem 3: There exists a new field k such that exactly three negative prime discriminants      divide d_k, the 4-rank of C_k is equal to 1, and d_k is congruent to 4 mod 8.

Proof: We first note that we can only apply Lemma 10 to fields that satisfy Rédei sub-type matrices 60e, f, g, i, since we have (q₂q₃q₄q₅/2) = -1 for these matrices and (q₂q₃q₄q₅/2) = 1 for the other six Rédei sub-type matrices listed above. We establish our theorem by applying Lemmas 10 and 14 to the field k = Q(√-5.13.7.83) = Q(√-37765), which satisfies Rédei matrix 60g.  In accordance with Lemma 14 we see that M₃ = Q(√5.13.7.83) = Q(√37765) and that h(M₃) ≥ 8. From [4] we see that h(M₃) = 16, and since 5 and 13 are congruent to 5 mod 8,           7 is congruent to 7 mod 8, and 83 is congruent to 3 mod 8, we obtain that (5.13.7.83/2) = -1, and we observe that the conditions of Lemma 11 are satisfied. Consequently from Lemma 10 we can conclude that k has infinite 2-class field tower, and since k satisfies Rédei matrix 60g we see that k is a new field, which proves our theorem.

Remark 9: We expect that using the technique described in Theorem 3, one can readily find additional new fields k as above, and we leave this to the interested reader to explore. However, when d_k is congruent to 4 mod 8 and C_k has 4-rank equal to 1, the 4-rank of C_j for a corresponding field j obtained by using CRT and DPAP as in Lemma 12 may not be the same as the 4-rank of C_k. To see an example of this, let k = Q(√-3.5.7.29) = Q(√-3045) and                      j = Q(√-3.263.5.7.29) = Q(√-800835). We see from the above Rédei matrix 4-property that        C_k has 4-rank 1 and C_j has 4-rank 0, and we also see that the field j is obtained from the field k by CRT and DPAP as described in Lemma 12. In addition, we see from [20] that field k belongs to Case 59, field j belongs to Case 34 (see below) and it is not known if either of these fields have infinite 2-class field tower. However, if d_k is not congruent to 4 mod 8 then for all fields in the infinite collection of fields described in Lemma 12, the 4-ranks of C_k and C_j will be the same.  

     We next show that in the remaining case when the 4-rank of C_k is equal to 1, exactly three negative prime discriminants divide d_k, and d_k is not congruent to 4 mod 8, there are infinitely many new fields k that satisfy the 2-class field tower conjecture.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            

 

Case 4: Exactly Three Negative Prime Discriminants Dividing d_k

with 4-Rank of C_k Equal to 1 and d_k Not Congruent to 4 mod 8

     For the case when exactly three negative prime discriminants divide d_k, the 4-rank of C_k is equal to 1, and d_k is not congruent to 4 mod 8, we see from [20] that there are seven open Rédei sub-type matrices of 4-rank 1, which are listed as matrices 31, 33, 35, 43, 47, 48, and 49 in [20], for which we describe as follows, using the above description of Rédei matrices.

Matrix 31: a₁₂ =a₁₃ = a₁₅ = a₂₃ = a₂₄ = a₂₅ = a₃₄ = a₄₅ = 1, a₁₄ = a₃₅ = 0

Matrix 33: a₁₂ =a₁₃ = a₁₅ = a₂₃ = a₂₄ = a₂₅ = a₃₄ = a₃₅ = a₄₅ = 1, a₁₄ = 0

Matrix 35: a₁₂ =a₁₃ = a₁₄ = a₁₅ = a₂₃ = a₂₄ = a₂₅ = a₃₅ = a₄₅ = 1, a₃₄ = 0

Matrix 43: a₁₂ = a₁₄ = a₂₃ = a₂₄ = a₃₅ = a₄₅ = 1, a₁₃ = a₁₅ = a₂₅ = a₃₄ = 0

Matrix 47: a₁₂ = a₁₅ = a₂₃ = a₂₅ = a₃₄ = a₃₅ = a₄₅ = 1, a₁₃ = a₁₄ = a₂₄ = 0

Matrix 48: a₁₂ = a₁₅ = a₂₃ = a₂₄ = a₃₄ = a₃₅ = a₄₅ = 1, a₁₃ = a₁₄ = a₂₅ = 0

Matrix 49: a₁₂ = a₁₅ = a₂₃ = a₂₄ = a₂₅ = a₃₄ = a₃₅ = a₄₅ = 1, a₁₃ = a₁₄ = 0

   We now make use of this categorization to establish the following theorem:

Theorem 4: There are infinitely many new fields k such that exactly three negative prime discriminants divide d_k, the 4-rank of C_k is equal to 1, and d_k is not congruent to 4 mod 8.

Proof: Let k = Q(√-q₁q₂q₃p₁p₂) where q_1, q₂, and q₃ are distinct negative prime discriminants,    p₁ andp₂ are distinct positive prime discriminants, and d_k is not congruent to 4 mod 8.            We determine which of the above seven open Rédei sub-type matrices of 4-rank 1 are candidates for new fields that satisfy a Rédei sub-type matrix. From the Rédei matrix 4-rank property and genus theory we see that C_M,2 is isomorphic to Z/2Z x Z/2Z for Rédei sub-type matrices 31, 43, 47, and 48, for all corresponding real quadratic number fields M to k in Theorem 11, since the ranks of these Rédei matrices are all 3. Therefore we are not able to use Theorem 11 to obtain new fields that satisfy these Rédei sub-type matrices. For Rédei sub-type matrix 49, we see from the Rédei matrix 4-rank property and genus theory that C_M,2 is isomorphic to Z/2Z x Z/2Z for M = Q(√q₁q₂p₁p₂) and M = Q(√q₁q₃p₁p₂), and that C_M has 4-rank 1 for M = Q(√q₂q₃p₁p₂). However, (q₂q₃p₁p₂/q₁) = 1 and therefore the Kronecker symbol requirement of Lemma 10 is not satisfied for fields that satisfy Rédei sub-type matrix 49. For Rédei sub-type matrices 33 and 35, although we obtain using the above methods that C_M,2 is isomorphic to Z/2Z x Z/2Z for two of the three possible fields M in each type, for Rédei sub-type matrix 33 with M = Q(√q₂q₃p₁p₂)    we obtain that C_M has 4-rank 1 and (q₂q₃p₁p₂/q₁) = -1, and for Rédei sub-type matrix 35 with          M = Q(√q₁q₂p₁p₂) we obtain that C_M has 4-rank 1 and (q₁q₂p₁p₂/q₃) = -1. We illustrate this result for M = Q(√q₂q₃p₁p₂ in matrix 33, and we leave the remaining details for the reader to check.                                                                                             

 

     Given the above Rédei matrix entries for matrix 33 we see that the Rédei matrix for

M = Q(√q₁q₂p₁p₂) is the following:

R_M = [0   1   1   1        

           0   1   1   1

           1   1   1   1

           1   1   1   1]

It therefore follows that R­_M has rank 2 and by the Rédei matrix 4-rank property we are able to conclude that C_M has 4-rank 1. From the listing of the Rédei matrix entries for matrix 33 we seethat (q₂q₃p₁p₂/q₁) = -1.  

     We thus see that Rédei sub-type matrices 33 and 35 are the only possible candidates for new fields that satisfy a Rédei sub-type matrix of 4-rank 1 in the case that we are now considering.                                                           

From [11] we see that for the field M= Q(√19.11.13.41) = Q(√111397) we have h(M) = 16, and we are able to use this field M to obtain an imaginary quadratic number field k that satisfies Rédei sub-type matrix 33 and the conditions of Lemma 10 (see Example 8 and Remark 10 below). Consequently from Lemmas 10 and 12, and Remark 8, we are able to conclude that there are infinitely many new fields that satisfy the conditions of our theorem.          

 

Example 8: Let k = Q(√-19.11.191.13.41) = Q(√-21276827); q­₃ = 191, (19/191) = (11/191) = (41/191) = -1, (13/191) = 1, (19.11.13.41/191) = -1, h(Q(√19.11.13.41)) = 16; we see that

k satisfies Rédei sub-type matrix 33, and consequently from Lemma 10 we obtain that k is a

new field with infinite 2-class field tower, and from Lemma 12 and Remark 8 we conclude that there are infinitely many new fields with infinite 2-class field tower that satisfies the conditions of Theorem 2.

 

Remark 10: We expect that by the use of [4] and Lemmas 10 and 12, one can readily obtain infinitely many new fields that satisfy Rédei sub-type matrix 35, and we leave this task to the interested reader. We note that if M is a real quadratic number field corresponding to a field k as in Lemma 11, such that h(M) ≥ 16 and the Kronecker symbols of the primes dividing d_M are consistent with the entries of Rédei sub-type matrix 35 (resp. matrix 33), then by CRT and DPAP     (see the proof of Lemma 13) we know that there exist infinitely many fields j with infinite          2-class field tower that satisfy Rédei sub-type matrix 35 (resp. 33), and since d_k is not congruent to 4 mod 8 we see from Remark 8 that the 4-rank of C_j is equal to 1 for all fields j in this infinite collection and consequently we would obtain infinitely many new fields.

 

     We summarize our results thus far for obtaining new fields k that satisfy the 2-class field tower conjecture in the following table, where once again k is an imaginary quadratic number fields such that rank C_k,2 = 4, “Negative Prime Discriminants” denotes the exact number of negative prime discriminants dividing d_k, “Congruency” denotes whether or not d_k is congruent to 4 mod 8, “Examples” denotes the examples we have supplied, and q₃ denotes a prime congruent to 3 mod 4 that has the same kronecker symbols with the primes 19, 11, 13, and 41 as does the prime 191, as described in the proof of Lemma 12.    

 

 

Table 2: New Fields that Satisfy the 2-Class Field Tower Conjecture

4-rank of C­_k    Negative Prime Discrminants       Congruency                       Examples                            

       2                                   3                                      4 mod 8                k = Q(√-5.13.7.827)                                                  

                                                                                                                   k = Q(√-5.13.47.827)                                            

                                                                                                                   k = Q(√-5.29.47.827)                                            

                                                                                                                      k = Q(√-5.29.47.43)                                            

       1                                   5                                      4 mod 8                k = Q(√-23.19.67.3)                                            

                                                                                                                    k = Q(√-11.7.167.79)                                            

                                                                                                                    k = Q(√-23.11.19.103)                                            

         1                                   3                                      4 mod 8                   k = Q(√-5.13.7.83)

       1                                  3                                  not 4 mod 8         k = Q(√-19.11.191.13.41)

                                                                                                                 Infinitely Many Fields

                                                                                                                        of the form

                                                                                                                 k = Q(√-19.11.q₃.13.41)

 

     We next examine the case when C_k has 4-rank 0 instead of 4-rank 1 (i.e. C_k,2 is elementary), and exactly three or five negative prime discriminants divide d_k for the fields k that we are considering (i.e. once again k imaginary quadratic with rank C_k,2 = 4), to determine if there are new fields k that satisfy the conditions of Lemma 10. We begin with the five negative prime discriminant case.

 

 

 

Case 5: Five Negative Prime Discriminants Dividing d_k                                                          

with 4-Rank of C_k Equal to 0

   To begin with, we demonstrate that there are no new fields k that satisfy the conditions of Lemma 10 for the five negative prime discriminant case when the 4-rank of C_k is equal to 0.     We first obtain from [19] that there are two open Rédei sub-type matrices such that d_k is          not congruent to 4 mod 8 (Rédei matrices k and l in [19]), which are given in [19] as follows

(we designate these matrices as “A” and “B” for ease of notation):

A = [1   1   1   0   1          B = [0   1   1   0   0      

       0   0   1   1   0                      0   0   1   1   0            

       0   0   0   1   1                      0   0   0   1   1                    

       1   0   0   0   1                      1   0   0   0   1

       0   1   0   0   1]                  1   1   0   0   0]

     The following lemma shows that if k satisfies Rédei matrix A or B then the required condition in Lemma 11 that h(M)| ≥ 16 is not satisfied.

Lemma 15: Let k satisfy Rédei matrices A or B (as described above), and let M be any of the corresponding real quadratic number fields given in Lemma 11. Then C_M,2 is isomorphic to Z/2Z x Z/2Z.

Proof: Let k = Q(√-q₁q₂q₃q₄q₅) and M = Q(√q_iq_jq_kq_l), q_i congruent to 3 mod 4, 1 ≤ i ≤ 5,               q_i all distinct and {i, j, k, l} C {1, 2, 3, 4, 5}. If k satisfies Matrix A then we have the following Kronecker symbols: (-q₁/q₂) = (-q₁/q₃) = (-q₂/q₃) = (-q₂/q₄) = (-q₃/q₄) = (-q₃/q₅) = (-q₁/q₅) =      (-q₄/q₅) = -1, (-q₁/q₄) = (-q₂/q₅) = 1. Consequently we have the following possible              Rédei matrices for M:

M₁ = Q(√q₁q₂q₃q₄)      M₂ = Q(√q₁q₂q₃q₅)      M₃ = Q(√q₁q₂q₄q₅)      M₄ = Q(√q₃q₂q₄q₅)

   [1   1   1   0                 [0   1   1   1               [0   1   0   1                 [1   1   0   1    

   0   0   1   1                   0   0   1   0                 0   0   1   0                   0   1   1   1

   0   0   0  1                   0    0   0   1                 0   0   1   1                   1   0   1   1

   1   1   0   0]                   0    1   0   0]               0   1   0   0]                 0   0   0   1]

M₅ = Q(√q₂q₃q₄q₅)    

   [1   1   1   0                  

   0   1   1   1                        

   0   0   0   1

   1   0   0   0]  

     Since all matrices M₁,M₂, M₃, M₄, M₅ have rank 3, we see from genus theory and the     Rédei matrix 4-rank property that the 4-rank of M is equal to 0 and thus C_M,2 is isomorphic to     Z/2Z x Z/2Z. In a similar way, it can be shown that if k satisfies Matrix B then C_M,2 is isomorphic to Z/2Z x Z/2Z (we once again leave the details to the reader) and consequently our lemma is proved.    

     For the case when five negative prime discriminants divide d_k, C_k,2 is elementary, and            d_k is congruent to 4 mod 8, we see from [19] that there are three open Rédei sub-type matrices which we designate as matrices C, D₁, and D₂ (derived from matrices o and p in [19]), which we give as follows, where once again the * signifies a choice of 0 or 1:

C = [*   1   1   1   1           D₁ = [*   1   1   1   1          D₂ = [*   1   1   1   1    

       *   0   1   0   1                      *   0   1   1   0                   *   0   1   1   0

       *   0   0   1   1                      1   0   0   1   1                     0   0   0   1   1

       *   1   0   0   1                      1   0   0   1   1                     0   0   0   1   1

       1   0   0   0   0]                     *   1   0   0   1]                   *   1   0   0   1]

     The following lemma shows that if k satisfies Rédei matrices C, D₁, or D₂, then k does not satisfy the conditions of Lemma 10.

Lemma 16: Let k = Q(√-q₂q₃q₄q₅) with q_i congruent to 3 mod 4, 2 ≤ i ≤ 5, q_i all distinct,              let k satisfy Rédei matrices C, D₁, or D₂ (as described above), and let M be any of the corresponding real quadratic number field given in Lemma 5. Then either C_M,2 is isomorphic to Z/2Z x Z/2Z, or M =Q(√q_jq_jq_k) with i, j, k distinct, {i, j, k} C {2, 3, 4, 5} = {i, j, k, l}, and (q_jq_jq_k/q_l) = 1.

Proof: If k satisfies Matrix C then we have the following Kronecker symbols: (-q₂/q₃) = (-q₃/q₄) = (-q₃/q₅) = (-q₄/q₅) = (-q₂/q₄) = (-q₂/q₅) = -1, q₅ congruent to 7 mod 8. If M = Q(√q₂q₃q₄q₅) then we see that M has rank 3 and therefore the 4-rank of C_M is equal to 0 and C_M,2 is isomorphic to Z/2Z x Z/2Z . If M is not equal to Q(√q₂q₃q₄q₅) then M has the format given in the lemma, and we see that (q₂q₃q₄/q₅) = (q₂q₃q₅/q₄) = (q₂q₄q₅/q₃) = (q₃q₄q₅/q₂) = 1, which establishes our lemma when k satisfies Matrix C. If k satisfies Matrices D₁ or D₂ then we have the following Kronecker symbols: (-q₂/q₃) = (-q₃/q₄) = (-q₃/q₅) = (-q₄/q₅) = (-q₂/q₄) = -1, (-q₂/q₅) = 1.              If M = Q(√q₂q₃q₄q₅), Q(√q₂q₃q₄), Q(√q₂q₃q₅), or Q(√q₃q₄q₅), then once again we see that M has rank 3 and thus the 4-rank of C_M is equal to 0 and C_M,2 is isomorphic to Z/2Z x Z/2Z .                 If M = Q(√q₂q₄q₅) then (q₂q₄q₅/q₃) = 1, and if M = Q(√q₃q₄q₅) then (q₃q₄q₅/q₂) = 1; consequently our lemma has been proved.

     We therefore are able to conclude from Lemmas 15 and 16 that Lemma 10 is not applicable in the case when five negative prime discriminants divide d_k and C_k,2 is elementary, and consequently by our present techniques we are not able to formulate any new fields that have infinite 2-class field tower in this case. We now examine the case when exactly three negative prime discriminants divide d_k and C_k,2 is elementary.

 

Case 6: Exactly Three Negative Prime Discriminants Dividing d_k

with 4-Rank of C_k Equal to 0

     We demonstrate that there are also no new fields k that satisfy the conditions of Lemma 10 when exactly three negative prime discriminants divide d_k and the 4-rank of C_k is equal to 0. We first obtain from [20] that there are seven open Rédei sub-type matrices such that d_k is not congruent to 4 mod 8 (Rédei matrices 16, 28, 30, 32, 34, 49 in [20]), which are given in [20] as follows, where we designate Cases 34a and 34b to represent the two possible open Rédei      sub-type matrices for Case 34, where once again the * signifies a choice of 0 or 1:

#16 = [1   1   1   1   1           #28 = [1   1   1   0   1           #30 = [1   1   1   0   1    

           0   0   1   0   1                        0   1   1   1   1                     0   0   1   1   0

           0   0   1   1   0                        0     0   1   0   1                     0   0   0   1   1

           1   0   1  1   1                        0   1   0   0  1                       0   1   1   1   1

           0   1   0   1   0]                     1   1   1   1    0]                      1   0   1   1   1]

#32 = [0   1   1   1   1          #34a = [0   1   1   1   1           #34b = [0   1   1   1   1                                                

          0   0   1   1   0                          0   0   1   0   1                        0   0   1   0   1

          0   0   0   1   1                          0   0   0   1   1                        0   0   0   1   1

          0   1   1   1   1                          1   0   1   0   0                          1   0   1   1   1

          1   1   0   1   1]                       1   1   1   0   1]                        1   1   1   1   0]

#49 = [0   1   0   0   1          

          0     1   1   1   1                              

          1     0   1   1   1

          0     1   1   0   0                              

          1     1   1   0   1]

The following lemma shows that if k satisfies any of the above seven Rédei matrices then           k does not satisfy the conditions of Lemma 10.

Lemma 17: Let k = Q(√-q₁q₂q₃p₁p₂) where q_1, q₂, and q₃ are distinct negative prime discriminants, p₁ andp₂ are distinct positive prime discriminants, and d_k is not congruent to         4 mod 8. Let k satisfy Rédei matrices 16, 28, 30, 32, 34a, 34b, or 49, and let M be any of the

corresponding real quadratic number field given in Lemma 10. Then either C_M,2 is isomorphic to

Z/2Z x Z/2Z, or (without loss of generality) M = Q(√q₁q₂p₁p₂) with (q₁q₂p₁p₂/q₃) = 1.

Proof: We illustrate the proof by examining matrices 16 and 34a. For matrix 16 we have the following three possible matrices for M:

M₁ = [1     1     0     0               M₂ = [1     1     1     0           M₃ = [1     1     1     0          

           0     0     1     0                           0     0     0     1                     0     1     1     0

           0     1     0     1                           1     0     0     1                     1     1     1     1

           1     0     1     0]                         0     1     1     0]                    0     0     1     1]

Since the rank of each of the above three matrices is 3, we see from genus theory and the     Rédei matrix 4-rank property that C_M,2 is isomorphic to Z/2Z x Z/2Z for M as stated in the lemma. It can readily be determined that this same result holds for matrices 28, 30, and 32,     and we once again leave the details for the reader to check. For matrix 34a we have the following three possible matrices for M, where M₃ = Q(√q₁q₃p₁p₂):

M₁ = [1     1     0     1               M₂ = [0     1     1     1           M₃ = [0     1     1     1          

           0     0     1     1                           0     0     0     1                     0     1     1     1

          0     1     1     0                           1     0     1     0                     1     1     0     0

           1     1     0     0]                         1     1     0     0]                   1     1     0     0]

Since matrices M₁ and M₂ have rank 3 and matrix M₃ has rank 2, we see as above that C_M,2 is isomorphic to Z/2Z x Z/2Z when M = M₁ or M₂, and C_M has 4-rank 1 if M = M₃. However, since (q₁q₃p₁p₂/q₂) = 1 our lemma is established for matrix 34a. Similarly it can be shown that for matrices 34b and 49 we obtain that C_M,2 is isomorphic to Z/2Z x Z/2Z when M = M₁ or M₂, and C_M has 4-rank 1 if M = M₃ with (q₁q₃p₁p₂/q₂) = 1 (without loss of generality; we once again leave the details to the reader to check), and consequently our lemma has been proved.

     Finally, we examine the case when exactly three negative prime discriminants divide d_k,    C_k,2 is elementary, and d_k is congruent to 4 mod 8. We observe from [20] that there are 26 open Rédei sub-type matrices, which belong to Cases 56, 57, 58, and 59 in [20]. As an illustration we list the eight open Rédei sub-type matrices for Case 59 as follows, where a₁₂ = a₁₃ = a₂₃ = a₂₄ = a₃₄ = a₃₅ = 1, a₁₄ = a₁₅ = a₂₅ = 0:

a) a₂₁ = a₃₁ = a₄₁ = a₅₁ = 1, a­₄₅ = 0

b) a₂₁ = a₃₁ = a₅₁ = 1, a­₄₅ = a­₄₁ = 0

c) a₄₁ = a₅₁ = 1, a­₄₅ = a­₂₁ = a₃₁ = 0

d) a­₄₅ = a­₄₁ = a₅₁ = 1, a₂₁ = a₃₁ = 0

e) a­₄₅ = a­₃₁ = a₄₁ = a₅₁ = 1, a₂₁ = 0

f) a­₄₅ = a­₂₁ = a₃₁ = a₄₁ = 1, a₅₁ = 0

g) a­₄₅ = a­₅₁ = 1, a₂₁ = a₃₁ = a₄₁ = 0

h) a­₄₅ = a₃₁ = a­₅₁ = 1, a₂₁ = a₄₁ = 0

     Using similar techniques to what we have described above, it can readily be shown that C_M,2 is isomorphic to Z/2Z x Z/2Z for all corresponding real quadratic number fields M for each of the above 26 open Rédei sub-type matrices (we once again leave the details to the reader to verify) and consequently we see that there are no new fields k that satisfy the conditions of Lemma 10 when exactly three negative prime discriminants or five negative prime discriminants divide d_k and the 4-rank of C_k is equal to 0.

 

Open Questions on the 2-Class Field Tower Conjecture

     Although we have found new fields k, where k is an imaginary quadratic number field with rank C_k,2 = 4 and C_k having 4-rank 1 or 2 that satisfy the 2-class field tower conjecture, and infinitely many new fields in the 4-rank 1 case, there are still many open questions. We state the following five open questions.

Question 1: Do there exist infinitely many new fields k as above such that rank C_k,2 = 4 and                     C_k has 4-rank 2?

Question 2: Do there exist new fields k such that rank C_k,2 = 4 and C_k has 4-rank 2 for case d    as described in the 4-rank of C_k equal to 2 section? (see Question 4)

Question 3: Do there exist infinitely many new fields k such that rank C_k,2 = 4 and C_k has          4-rank 1 in the cases when d_k is congruent to 4 mod 8 and exactly three negative prime discrimants or five negative prime discriminants divide d_k?

Question 4: Do there exist new fields k as above that do not satisfy the conditions of Lemma 11, such that k has infinite 2-class field tower? Perhaps such a field k can be obtained by satisfying the conditions of Lemma 8 (cf. Prop. 3.4 in [14]), but we have not found any such fields k that satisfy the conditions of Lemma 8 without also satisfying the conditions of Lemmas 4, 5, or 7. We note that any new field satisfying the conditions given in Question 2 cannot be obtained from Lemma 10 (cf. Remark 8).  

Question 5: Do there exist new imaginary quadratic number fields k with rank C_k,2 = 4 that satisfy the conditions of Lemma 10 and consequently have infinite 2-class field tower in the case when the 4-rank of C_k is 0, for which exactly three or five negative prime discriminants divide d_k and for which d_k is congruent to 4 mod 8 or not congruent to 4 mod 8?  

 

Acknowledgements

     I would like to thank Chip Snyder for a number of productive discussions concerning the results of this paper, and Yukata Sueyoshi for his helpful personal communication conveying to me that my field in Example 1 is indeed a new field, as well as for sending me his related 2010 paper.

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