Fields Institute Communications Volume 00, 0000

Q-adequate Bicyclic Bicubic Fields

Helen G. Grundman

Department of Mathematics Bryn Mawr College 101 N. Merion Ave. Bryn Mawr Pennsylvania 19010

Tara L. Smith

Department of Mathematical Sciences University of Cincinnati PO Box 210025 Cincinnati Ohio 45221-0025

Abstract. This paper characterizes those bicyclic bicubic extensions of the rational numbers that can arise as a maximal commutative subfield of a division algebra with center the rational numbers. These fields in turn determine all Galois 3-extensions of the rationals which can be maximal subfields of such division algebras.

1 Introduction

Let L be a finite extension of a field F. If there exists an F -central division algebra D containing L as a maximal commutative subfield, L is said to be F -adequate; otherwise L is F -deficient. In this paper we characterize those bicyclic bicubic extensions of Q which are Q-adequate. Recall that a bicyclic bicubic extension is a Galois extension L over Q with Gal(L/Q) ≅ Z/3Z × Z/3Z. By [4, Theorem 2.5], we know that for a global field F, the F -adequacy of a finite Galois p-extension L is determined by the F -adequacy of the maximal elementary abelian intermediate extension. By [6, Theorem 4.1], the maximal elementary abelian extension of a Q-adequate p-extension L is of degree p (in which case L is a cyclic extension of Q) or of degree p^{2}. As cyclic extensions of global fields are always adequate extensions

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. 11R32, 11S20, 12F10, 16K20.

The first author was supported in part by the Faculty Research Fund of Bryn Mawr College.

([6, Theorem 2.8]), the determination of Q-adequate bicyclic bicubic extensions of Q provides a complete characterization of those Galois 3-extensions of Q that are Q-adequate.

Essential to our analysis of the Q-adequacy of bicyclic bicubic extensions is the following result obtained by specializing [6, Propositions 2.1 & 2.5] to the bicyclic bicubic case. (Here L_{p} denotes the localization of L at p. If L = Q(θ_{1}, θ_{2}), where θ_{i} is a root of an irreducible cubic polynomial over Q, then L_{p} = Q_{p}(θ_{1}, θ_{2}).)

Theorem 1.1 A bicyclic bicubic extension L of Q is Q-adequate if and only if [L_{p} : Q_{p}] = 9 for two different rational primes p.

Before discussing the determination of these local degrees we give some background on the structure of cyclic cubic and bicyclic bicubic extensions of Q, fixing notation we will use throughout. We start with a general theorem on cyclic cubic fields.

Lemma 1.2 [1, Theorem 6.4.6] Each cyclic cubic field K with conductor f is given exactly once (up to isomorphism) in the following way: K = Q(θ) where q(θ) = 0 with q(x) ∈ Z[x] given as follows.

1. If 3|f, q(x) = x^{3} − \frac{f}{3} x− \frac{fa}{27}, where 4f = a^{2} +27b^{2}, a ≡ 6 (9), 3 |̸ b, a ≡ b (2), b > 0, and \frac{f}{9} is a product of distinct primes congruent to 1 modulo 3.

2. If 3 |̸ f, q(x) = x^{3} − x^{2} + \frac{1−f}{3} x − \frac{1−3f+fa}{27}, where 4f = a^{2} + 27b^{2}, a ≡ 2 (3), a ≡ b (2), b > 0, and f is a product of distinct primes congruent to 1 modulo 3.

In both cases, the discriminant of q(x) is b^{2}f^{2} and the discriminant of K is f^{2}.

Further, if t is the number of distinct primes dividing f, then there are exactly 2^{t−1} cyclic cubic fields of conductor f.

Now assume L is a bicyclic bicubic field with conductor f_{L}. Then L contains four cubic intermediate fields; we will denote them by K_{1} = Q(θ_{1}), K_{2} = Q(θ_{2}), K_{3} = Q(θ_{3}), K_{4} = Q(θ_{4}), where each θ_{i} is the root of an irreducible cubic polynomial q_{i}(x) as given in Lemma 1.2. For each i, we let f_{i} denote the conductor of K_{i} and let a_{i}, b_{i}, correspond to a, b, as in Lemma 1.2, and define α_{i} = \frac{a_{i}+3b_{i}\sqrt{−3}}{2}.

Let k = Q(\sqrt{−3}) = Q(ζ_{3}) where ζ_{3} is a primitive third root of unity. Then k(θ_{i}) is a pure cubic extension. Using the cubic formula, it is not difficult to show that k(θ_{i}) = k(\sqrt[3]{f_{i}α_{i}}) = k(\sqrt[3]{f_{i}\overline{α}_{i}}) where \overline{α}_{i} denotes the complex conjugate of α_{i}. Parry [5] uses this to prove the following two results which give useful information concerning the conductors of L and its cubic subfields.

Lemma 1.3 [5, Corollary 1] For any prime p, p divides either none or exactly three of the conductors f_{1}, f_{2}, f_{3}, f_{4}.

Lemma 1.4 [5, Theorem 4] Let p_{1}, p_{2}, …, p_{n} be the distinct prime divisors of

f_{1}f_{2} other than 3. Then f_{L} = \{ \begin{matrix}9p_{1} ⋯ p_{n} if 3|f_{1}f_{2} \\ p_{1} ⋯ p_{n} if 3 |̸ f_{1}f_{2} \end{matrix} .

In the following section, we will need Hensel’s Lemma repeatedly. We state here the version found in [3].

Lemma 1.5 (Hensel’s Lemma) Let F (x) ∈ Z_{p}[x] and let a_{0} ∈ Z_{p} with F (a_{0}) ≡ 0 (mod p) and F′(a_{0}) ≢ 0 (mod p). Then there exists a unique a ∈ Z_{p} such that F (a) = 0 and a ≡ a_{0} (mod p).

Q-adequate Bicyclic Bicubic Fields 3

2 Determining Local Degrees

In this section, we determine conditions under which [L_{p} : Q_{p}] = 9 for a given prime p. If L_{p}/Q_{p} is unramified, then Gal(L_{p}/Q_{p}) is a cyclic subgroup of Gal(L/Q) ≅ Z/3Z × Z/3Z, and so the extension is of degree at most 3. Thus we need only consider ramified extensions, and therefore we need only look at those primes p dividing f_{L}.

Proposition 2.1 Let L/Q be a bicyclic bicubic extension. Assume p |̸ f_{1} and p|f_{2}. The following are equivalent:

1. [L_{p} : Q_{p}] = 9.

2. [Q_{p}(θ_{i}) : Q_{p}] = 3 for all i.
3. [Q_{p}(θ_{1}) : Q_{p}] = 3.

Proof. That (1) implies (2) and (2) implies (3) are clear. To see (3) implies (1), observe that for i ≠ 1, p|f_{i}, and so [Q_{p}(θ_{i}) : Q_{p}] = 3. (Since the discriminant is not a unit, the extension must be proper.) Moreover, since f_{1} is a unit in Q_{p} but f_{2} is not, we cannot have Q_{p}(θ_{1}) = Q_{p}(θ_{2}). Thus if [Q_{p}(θ_{1}) : Q_{p}] = 3, we have two distinct cubic subfields in L_{p}, and so necessarily [L_{p} : Q_{p}] = 9. □

We first apply Proposition 2.1 to the special case of p = 3. Suppose 3|f_{L}. Then there is a unique cubic subfield K_{1} of L with 3 |̸ f_{1}. Using the notation in Lemma 1.2, we need to determine when [Q_{3}(θ_{1}) : Q_{3}] = 3. In Propositions 2.2 and 2.3 we prove that [L_{3} : Q_{3}] = 9 if and only if 3 |̸ b_{1}.

Proposition 2.2 If 3|b_{1}, then Q_{3}(θ_{1}) = Q_{3}, and [L_{3} : Q_{3}] ≤ 3.

Proof. From Lemma 1.2 we have q_{1}(x) = x^{3} − x^{2} + \frac{1−f_{1}}{3} x − \frac{1−3f_{1}+f_{1}a_{1}}{27} , where 4f_{1} = a_{1}^{2} + 27b_{1}^{2}, b_{1} ≡ 0 (3), a_{1} ≡ 2 (3), a_{1} ≡ b_{1} (2), f_{1} ≡ 1 (3). We will show that q_{1}(x) factors over Z/3Z as (x−n_{1})(x−n_{2})^{2}. Then, since q′_{1}(x) = 3x^{2}−2x+ \frac{1−f_{1}}{3} ≡ −2x + \frac{1−f_{1}}{3} (mod 3) has just one root, there exists a root of q_{1}(x) which is not a root of q′_{1}(x). Thus we can apply Hensel’s Lemma to show that q_{1}(x) has a root in Z_{3}, and therefore [(K_{1})_{3} : Q_{3}] = 1.

Let a_{1} = 2 + 3m. Reducing q_{1}(x) modulo 3, we have

q_{1}(x) = x^{3} − x^{2} + \frac{1 − f_{1}}{3} x − \frac{1 − 3f_{1} + f_{1}a_{1}}{27}
= x^{3} − x^{2} + \frac{4 − a_{1}^{2} − 27b_{1}^{2}}{3 · 4} x − \frac{4 − 3a_{1}^{2} − 81b_{1}^{2} + a_{1}^{3} + 27a_{1}b_{1}^{2}}{27 · 4}
≡ x^{3} − x^{2} − (4m + 3m^{2} + 9b_{1}^{2})x − (m^{2} + m^{3} − b_{1}^{2} + 3b_{1}^{2}m) (3)
≡ x^{3} − x^{2} − mx − (m^{2} + m^{3}) (3)
≡ x^{3} − x^{2} − mx − m^{2} − m (3)

Now we analyze this according to the class of m modulo 3. If m ≡ 0, q_{1}(x) ≡ x^{3} −x^{2} ≡ x^{2}(x−1). If m ≡ 1, q_{1}(x) ≡ x^{3} −x^{2} −x+1 ≡ (x−1)^{2}(x+1). If m ≡ −1, q_{1}(x) ≡ x^{3} −x^{2} +x ≡ (x+1)^{2}x. Thus in each case q_{1}(x) has a root modulo 3 which is not a root of q′_{1}(x) modulo 3, and so q_{1}(x) has a root in Q_{3} as claimed. □

Proposition 2.3 Keeping the notation above, and assuming 3 |̸ b_{1}, then q_{1}(x) is irreducible over Q_{3}. Thus [L_{3} : Q_{3}] = 9 in this case.

Proof. By Lemma 1.2, the discriminant of the polynomial q_{1}(x) is b_{1}^{2}f_{1}^{2}. Since 3 |̸ b_{1}^{2}f_{1}^{2}, 3 |̸ [O_{K_{1}} : Z[θ_{1}]], where O_{K_{1}} denotes the ring of algebraic integers in K_{1}. Therefore q_{1}(x) factors modulo 3 in the same way that the ideal (3) ⊆ O_{K_{1}} factors in O_{K_{1}}. Since 3 is not ramified, the polynomial q_{1}(x) does not factor as (x − n)^{3} modulo 3. Since Q(θ_{1})/Q is Galois, q_{1}(x) must then either factor as a product of three distinct linear factors or remain irreducible modulo 3. We will show the former cannot happen.

We have q_{1}(x) = x^{3} − x^{2} + \frac{1−f_{1}}{3} x − \frac{1−3f_{1}+f_{1}a_{1}}{27} . If 0 is a root modulo 3, then \frac{1−3f_{1}+f_{1}a_{1}}{27} ≡ 0 (3). If also 1 is a root, then \frac{1−f_{1}}{3} ≡ 0 (3). Then if in addition 2 is a root, we must have 2^{3} − 2^{2} = 4 ≡ 0 (3), which is impossible. Thus q_{1}(x) does not have three distinct roots modulo 3. Thus q_{1}(x) is irreducible modulo 3, and hence irreducible over Z_{3}, and so over Q_{3} as well. □

Next, we consider the determination of [L_{p} : Q_{p}] for p|f_{L}, p ≠ 3. Fix such a p and let K = Q(θ) be the unique cubic intermediate field of L such that p does not divide the discriminant of K.

By Lemma 1.2, p is congruent to 1 modulo 3. Again, we adopt the notation from that lemma.

Lemma 2.4 If p ≡ 1 (3), then −3 is a square in Q_{p}.

Proof. We know −3 is a square in Q_{p} if and only if −3 is a quadratic residue in Z/pZ. But for p ≡ 1 (3), quadratic reciprocity gives us

(\frac{−3}{p}) = (\frac{−1}{p}) (\frac{3}{p}) = (−1)^{\frac{p−1}{2}} (−1)^{\frac{p−1}{2}} (\frac{p}{3}) = (\frac{p}{3}) = 1,

so −3 ∈ (Q_{p})^{2} as desired. □

For each p ≡ 1 (3) fix r_{p} ∈ Z with r_{p}^{2} ≡ −3 (p) and 0 < r_{p} < \frac{p}{2}. Let \sqrt{−3} ∈ Q_{p} denote the corresponding square root of −3. Then α = \frac{a+3b \sqrt{−3}}{2} ∈ Q_{p} and K_{p} = Q_{p}(\sqrt[3]{fα}).

Theorem 2.5 [L_{p} : Q_{p}] = 9 if and only if (a^{2} + 27b^{2})(a + 3br_{p}) is not a cube in Z/pZ.

Proof. By Proposition 2.1 we have [L_{p} : Q_{p}] = 9 if and only if [K_{p} : Q_{p}] = 3. Since \sqrt{−3} ∈ Q_{p}, we have K_{p} = Q_{p}(\sqrt[3]{fα}), so [K_{p} : Q_{p}] = 3 if and only if x^{3} − fα has no root in Q_{p}. If x^{3} − fα has no root in Z_{p}/pZ_{p} ≅ Z/pZ, then it is also irreducible over Q_{p}. Conversely, suppose x^{3} − fα has a root in Z_{p}/pZ_{p}. Then since p |̸ fα, 0 is not a root of x^{3} − fα, and its derivative is 3x^{2}, which has only 0 as a root. Thus by Hensel’s Lemma, x^{3} − fα has a root in Q_{p}. Therefore [L_{p} : Q_{p}] = 9 if and only if fα is not a cube in Z_{p}/pZ_{p} if and only if 8fα = (a^{2} + 27b^{2})(a + 3b\sqrt{−3}) is not a cube in Z_{p}/pZ_{p}. Since \sqrt{−3} ≡ r_{p} modulo pZ_{p}, we get the desired result. □

3 Determining Q-Adequacy

In this section, we give criteria for bicyclic bicubic fields to be Q-adequate. We then apply our methods to the bicyclic fields of smallest discriminant, recording the results in Table 2.

As before, let L be a bicyclic bicubic extension of Q, L = K_{1}K_{2}, and assume 3 |̸ f_{1}. We assume a_{i}, b_{i} are given, i = 1, 2. By [5] the values of a_{i}, b_{i} for i = 3, 4 can be determined. Specifically, we have the following.

Lemma 3.1 [5, Corollary 2] Let δ = gcd(α_{1}, ᾱ_{2}), γ = gcd(α_{1}, α_{2}), where gcd denotes the greatest common divisor in O_{k}, the ring of integers of k = Q(\sqrt{−3}). Then there exist integers β_{1}, β_{2} ∈ O_{k} such that α_{1} = γδβ_{1}, α_{2} = γδ̄β_{2}. Further, α_{3} = γ̄β_{1}β_{2} or α_{3} = γβ̄_{1}β̄_{2}, and α_{4} = δ̄β_{1}β̄_{2} or α_{4} = δβ̄_{1}β_{2}.

Thus we can assume that, given L, we have the parameters a_{i}, b_{i}, for 1 ≤ i ≤ 4. Let

S = {p : p a rational prime, p ≠ 3, p|f_{L}, and [L_{p} : Q_{p}] = 9}.

Then by the results of Theorem 2.5, we have

S = {p : p ≠ 3, p|f_{L}, and (a_{i}^{2} + 27b_{i}^{2})(a_{i} + 3b_{i}r_{p}) ∉ (Z/pZ)^{3} for i such that p |̸ f_{i}}.

The definition of S together with Theorem 1.1, Proposition 2.2 and Proposition 2.3 give the following result.

Theorem 3.2 Let L/Q be a bicyclic bicubic extension, where 3 |̸ f_{1}.

1. Assume 3 |̸ f_{L}. Then L is Q-adequate if and only if |S| ≥ 2.

2. Assume 3|f_{L}. Then L is Q-adequate if and only if either (a) |S| ≥ 2 or

(b) 3 |̸ b_{1} and |S| ≥ 1.

We now consider the problem of determining Q-adequacy of a large number of bicyclic bicubic fields. We develop Theorem 3.5 which gives criteria for determining local degrees that can be applied without first computing the values of α_{3} and α_{4}. We begin by fixing some notation.

For each prime p ≡ 1 (3), consider the unique cyclic cubic field of conductor p. Let a_{[p]} and b_{[p]} be as in Lemma 1.2, with 4p = a_{[p]}^{2} + 27b_{[p]}^{2}. Fix a_{[3]} = −3 and b_{[3]} = 1. For p prime, p ≡ 0, 1 (3), let

α_{[p]} = \frac{a_{[p]} + 3b_{[p]}\sqrt{−3}}{2}.

Let τ_{1}, τ_{−1} : C → C be the identity and complex conjugation respectively. In what follows, p will always denote 3 or a prime congruent to 1 (3).

Lemma 3.3 Let K be a cyclic cubic field with conductor f. Let a, b be as in Lemma 1.2 and let α = \frac{a+3b\sqrt{−3}}{2}. Then for each distinct prime p|f, there exists c_{p} ∈ {1, −1} such that

α = ∏_{p|f} τ_{c_{p}}(α_{[p]}).

Proof. Let α_{0} = \frac{a_{0} + 3b_{0}\sqrt{−3}}{2} = ∏_{p|f} τ_{c_{p}}(α_{[p]}) for some c_{p} ∈ \{1, −1\}. Using

induction on the number of prime factors of f, it is easily verified that a_{0} and b_{0} satisfy the conditions in Lemma 1.2 except that b_{0} may be negative. In this case, \overline{α}_{0} can be used instead so that the conditions in Lemma 1.2 are satisfied.

Let t be the number of distinct primes dividing f. Then letting the c_{p} vary, there are 2^{t} different possible values for α_{0}, exactly half of which satisfy the conditions of Lemma 1.2. By that lemma, there are exactly 2^{t−1} cyclic cubic fields with conductor f, each determined by an α_{0} of this form. Hence a suitable choice of the c_{p} must yield the α that determines K. □

Recall that for each p ≡ 1 (3), we have r_{p} ∈ Z with r_{p}^{2} ≡ −3 (p) and 0 < r_{p} < \frac{p}{2}. Define ψ_{p} : O_{k} → Z/pZ by

ψ_{p} (\frac{u}{2} + \frac{v}{2} \sqrt{−3}) = \frac{u}{2} + \frac{v}{2} r_{p} ∈ Z/pZ.

Then ψ_{p} is easily seen to be a ring homomorphism. For s ∈ Q, q ≡ 1(3) and prime, and p ≡ 0, 1(3) and prime, define

χ(s, p, q) = \{ \begin{matrix}1 & if s ≠ ±1 \\ 0 & if s = ±1 and p = q \\ ψ_{q}(α_{[p]}^{2}\overline{α}_{[p]}) & if s = 1 and p ≠ q \\ ψ_{q}(α_{[p]}\overline{α}_{[p]}^{2}) & if s = −1 and p ≠ q \end{matrix} .

Lemma 3.4 Let K be a cyclic cubic field with a, b, α, and c_{p} as above. Then for q |̸ f with q ≡ 1 (3) and prime, (a^{2} + 27b^{2})(a + 3br_{q}) is a cube in Z/qZ if and only if ∏_{p|f} χ(c_{p}, p, q) is a cube in Z/qZ.

Proof.

∏_{p|f} χ(c_{p}, p, q) = ∏_{p|f} ψ_{q}(τ_{c_{p}}(α_{[p]}^{2}\overline{α}_{[p]}))
= ψ_{q}( ∏_{p|f} (α_{[p]}^{2}\overline{α}_{[p]}))
= ψ_{q}(( ∏_{p|f} τ_{c_{p}}(α_{[p]}))^{2}( ∏_{p|f} τ_{−c_{p}}(α_{[p]})))
= ψ_{q}(α^{2}\overline{α})
= (a^{2} + 27b^{2})(a + 3br_{q})/8.
□

Let L = K_{1}K_{2} be a bicyclic bicubic field with K_{1} and K_{2} distinct cyclic cubic subfields. Assume without loss of generality that 3 |̸ f_{1}. Using Lemma 3.3, for i = 1, 2, let c_{p,i} ∈ {1, −1} such that α_{i} = ∏_{p|f_{i}} τ_{c_{p,i}}(α_{[p]}). Then it follows from Lemma 3.1 that

α_{3} or \overline{α}_{3} = ∏_{\begin{matrix}p^{2}|f_{1}f_{2} \\ c_{p,1} = c_{p,2} \end{matrix}} τ_{−c_{p,1}}(α_{[p]}) ∏_{\begin{matrix}p|f_{1} \\ p |̸ f_{2} \end{matrix}} τ_{c_{p,1}}(α_{[p]}) ∏_{\begin{matrix}p |̸ f_{1} \\ p|f_{2} \end{matrix}} τ_{c_{p,2}}(α_{[p]})

α_{4} or \overline{α}_{4} = ∏_{\begin{matrix}p^{2}|f_{1}f_{2} \\ c_{p,1} ≠ c_{p,2} \end{matrix}} τ_{c_{p,2}}(α_{[p]}) ∏_{\begin{matrix}p|f_{1} \\ p |̸ f_{2} \end{matrix}} τ_{c_{p,1}}(α_{[p]}) ∏_{\begin{matrix}p |̸ f_{1} \\ p|f_{2} \end{matrix}} τ_{−c_{p,2}}(α_{[p]})

From this we can determine the values of c_{p,3} and c_{p,4}.

For 1 ≤ i ≤ 4 and p prime with p |̸ f_{i}, define c_{p,i} = 0.

Theorem 3.5 For q|f_{L}, q ≡ 1 (3) and prime, [L_{q} : Q_{q}] = 9 if and only if one of the following is not a cube in Z/qZ.

X_{1} = ∏_{p} χ(c_{p,1}, p, q)
X_{2} = ∏_{p} χ(c_{p,2}, p, q)
X_{3} = ∏_{p} χ ((c_{p,1} + c_{p,2})(1 − \frac{3}{2}c_{p,1}c_{p,2}), p, q)
X_{4} = ∏_{p} χ ((c_{p,1} − c_{p,2})(1 + \frac{3}{2}c_{p,1}c_{p,2}), p, q)

For each q, all but one of these products is zero in Z/qZ.

Proof. Using Theorem 2.5 and Lemma 3.4, it suffices to prove that if q |̸ f_{i}, for some i, then X_{i} is a cube in Z/qZ if and only if ∏_{p} χ(c_{p,i}, p, q) is a cube in Z/qZ, and if q|f_{i}, then X_{i} = 0.

For p |̸ f_{i}, we have c_{p,i} = 0, and so χ(c_{p,i}, p, q) = 1. Hence for i = 1, 2, X_{i} = ∏_{p|f_{i}} χ(c_{p,i}, p, q) as required.

Now for each p,

(c_{p,1} + c_{p,2})(1 − \frac{3}{2}c_{p,1}c_{p,2}) = \{ \begin{matrix}−c_{p,1} & when & p^{2}|f_{1}f_{2} and c_{p_{,}1} = c_{p,2} \\ 0 & when & p^{2}|f_{1}f_{2} and c_{p,1} ≠ c_{p,2} \\ c_{p,1} & when & p|f_{1} and p |̸ f_{2} \\ c_{p,2} & when & p |̸ f_{1} and p|f_{2} \\ 0 & when & p |̸ f_{1} and p |̸ f_{2} \end{matrix}

= εc_{p,3}

where ε = ±1 is independent of p. Therefore X_{3} = ∏_{p} χ (εc_{p,3}, p, q) = ∏_{p|f_{3}} χ (εc_{p,3}, p, q) which is a cube if and only if ∏_{p|f_{3}} χ (c_{p,3}, p, q) is a cube.

Similarly for each p,

(c_{p,1} − c_{p,2})(1 + \frac{3}{2}c_{p,1}c_{p,2}) = \{ \begin{matrix}0 & when & p^{2}|f_{1}f_{2} and c_{p,1} = c_{p2} \\ −c_{p,1} & when & p^{2}|f_{1}f_{2} and c_{p1} ≠ c_{p,2} \\ c_{p,1} & when & p|f_{1} and p |̸ f_{2} \\ −c_{p,2} & when & p |̸ f_{1} and p|f_{2} \\ 0 & when & p |̸ f_{1} and p |̸ f_{2} \end{matrix}

= εc_{p,4}

where ε = ±1 is independent of p. And the result follows.

Finally, if q|f_{i}, then χ(c_{q,i}, q, q) = 0 will be a factor in the product and hence X_{i} = 0. □

Theorem 3.6 Let L/Q be a bicyclic bicubic extension. Using the notation above, L is Q-adequate if and only if either

1. there exist two primes p, p|f_{L}, p ≡ 1 (mod 3), such that, for some i, X_{i} is not a cube modulo p or

2. there exists a prime p, p|f_{L}, p ≡ 1 (mod 3), such that, for some i, X_{i} is not a cube modulo p, 3|f_{L}, and for the cubic subfield of conductor not divisible by 3, 3 |̸ b.

Proof. This follows immediately from Theorem 1.1, Propositions 2.2 and 2.3, and Theorems 3.2 and 3.5. □

We end with two tables. The first gives cyclic cubic fields of smallest discriminant, the conductor f, and the values of a and b for each. The second gives the bicyclic bicubic fields of smallest discriminant as determined in [2], the conductor f_{L}, two cubic subfields, whether or not the field is Q-adequate, and if it is, two primes p for which [K_{p} : Q_{p}] = 9.

Field f  a b Field f  a b Field  f  a b
F_{7}  7  -1 1 F_{97}  97 -19 1 F_{193} 193 23 3
F_{9}  9  -3 1 F_{103} 103 -13 3 F_{199} 199 11 5
F_{13}  13  5 1 F_{109} 109 2 4 F_{211} 211 -13 5
F_{19}  19 -7 1 _{1}F_{117} 9 · 13 6 4 _{1}F_{217} 7 · 31 29 1
F_{31}  31 -4 2 _{2}F_{117} 9 · 13 -21 1 _{2}F_{217} 7 · 31 -25 3
F_{37}  37 11 1 F_{127} 127 20 2 F_{223} 223 -28 2
F_{43} 43 8 2 _{1}F_{133} 7 · 19 17 3 F_{229} 229 -22 4
F_{61}  61 -1 3 _{2}F_{133} 7 · 19 -10 4 F_{241} 241 17 5
1F_{63} 7 · 9 15 1 F_{139} 139 23 1 _{1}F_{247} 13 · 19 -4 6
2F_{63} 7 · 9 -12 2 F_{151} 151 -19 3 _{2}F_{247} 13 · 19 -31 1
F_{67} 67 5 3 F_{157} 157 14 4 F_{271} 271 29 3
F_{73}  73 -7 3 F_{163} 163 -25 1 F_{277} 277 26 4
F_{79}  79 17 1 _{1}F_{171} 9 · 19 24 2 _{1}F_{279} 9 · 31 33 1
1F_{91} 7 · 3 11 13 _{2}F_{171} 9 · 19 -3 5 _{2}F_{279} 9 · 31 -21 5
2F_{91} 7 · 13 -16 2 F_{181} 181 -7 5 F_{283} 283 32 2

Figure 1 Cyclic Cubic Fields

f_{L} K_{1} K_{2} Q-ad? p  f_{L} K_{1} K_{2} Q-ad? p
63 F_{7} F_{9}  yes  3, 7 819 F_{7}  _{1}F_{117} yes  3, 7
91 F_{7} F_{13}  no  819 F_{7}  _{2}F_{117} no
117 F_{9} F_{13} yes 3,13 819 F_{9}  _{1}F_{91} yes 7,13
133 F_{7} F_{19}  no  819 F_{9}  _{2}F_{91} yes  3,7
171 F_{9} F_{19}  no  819 F_{13}  _{1}F_{63} yes  3,7
217 F_{7} F_{31} yes 7, 31 819 F_{13}  _{2}F_{63} yes  3,7
247 F_{13} F_{19} yes 13,19 819 _{1}F_{63} _{1}F_{91} yes 7,13
259 F_{7} F_{37} yes 7, 37 819 _{1}F_{63} _{2}F_{91} yes 3,13
279 F_{9} F_{31} yes 3, 31 819 _{2}F_{63} _{1}F_{91}  no
301 F_{7} F_{43}  no  819 _{2}F_{63} _{2}F_{91} yes 3,13
333 F_{9} F_{37}  no  871 F_{13} F_{67}  yes 13,67
387 F_{9} F_{43} yes 3, 43 873 F_{9} F_{97}  yes 3,97
403 F_{13} F_{31} no 889 F_{7} F_{127} no
427 F_{7} F_{61} yes 7,61 927 F_{9} F_{103}  no
469 F_{7} F_{67} yes 7,67 949 F_{13} F_{73}  no
481 F_{13} F_{37} yes 13,37 973 F_{7} F_{139} yes 7,139
511 F_{7} F_{73}  no  981 F_{9} F_{109}  no
549 F_{9} F_{61}  no  1057 F_{7} F_{151} yes 7,151
553 F_{7} F_{79} yes 7,79 1099 F_{7} F_{157}  no
559 F_{13} F_{43} yes 13,43 1141 F_{7} F_{163} yes 7,163
589 F_{19} F_{31}  no  1143 F_{9} F_{127}  no
603 F_{9} F_{67}  no  1147 F_{31} F_{37}  no
657 F_{9} F_{73}  no  1159 F_{19} F_{61}  yes 19,61
679 F_{7} F_{97}  no  1197 F_{7}  _{1}F_{171} yes  3,7
703 F_{19} F_{37}  no  1197 F_{7}  _{2}F_{171} yes  3,7
711 F_{9} F_{79} yes 3,79 1197 F_{9}  _{1}F_{133} no
721 F_{7} F_{103} yes 7,103 1197 F_{9} _{2}F_{133} yes 3,7
763 F_{7} F_{109} yes 7,109 1197 F_{19}  _{1}F_{63} yes 3,19
793 F_{13} F_{61} yes 13,61 1197 F_{19}  _{2}F_{63} yes 3,19
817 F_{19} F_{43} yes 19,43 1197 _{1}F_{63} _{1}F_{133} yes 7,19

Figure 2 Q-adequacy of Bicyclic Bicubic Fields

f_{L} K_{1} K_{2} Q-ad?  p  f_{L} K_{1} K_{2} Q-ad?  p
1197 _{1}F_{63} _{2}F_{133} yes  3, 7 1729 _{1}F_{91} _{1}F_{133} yes  7,13
1197 _{2}F_{63} _{1}F_{133} yes  7, 19 1729 _{1}F_{91} _{2}F_{133} yes  7,13
1197 _{2}F_{63} _{2}F_{133} yes  3,7 1729 _{2}F_{91} _{1}F_{133} yes  7,13
1251 F_{9} F_{139} yes 3,139 1729 _{2}F_{91} _{2}F_{133} yes  7,13
1261 F_{13} F_{97}  yes 13,97 1737 F_{9} F_{193}  no
1267 F_{7} F_{1811} no  1791 F_{9} F_{199}  no
1273 F_{19} F_{67}  yes 19,67 1807 F_{13} F_{139} yes 13,139
1333 F_{31} F_{43}  yes 31,43 1843 F_{19} F_{97}  no
1339 F_{13} F_{103}  no  1891 F_{31} F_{61}  no
1351 F_{7} F_{193} yes 7,193 1897 F_{7} F_{271} yes 7,271
1359 F_{9} F_{151} no 1899 F_{9} F_{211} yes 3,211
1387 F_{19} F_{73}  yes 19,73 1939 F_{7} F_{277} yes 7,277
1393 F_{7} F_{199} yes 7,199 1953 F_{7}  _{1}F_{279} yes  3,7
1413 F_{9} F_{157} yes 3,157 1953 F_{7}  _{2}F_{279} yes  3,31
1417 F_{13} F_{109}  no  1953 F_{9}  _{1}F_{217} yes  3,7
1467 F_{9} F_{163}  no  1953 F_{9}  _{2}F_{217} yes  7,31
1477 F_{13} F_{37}  yes 13,37 1953 F_{31}  _{1}F_{63} yes  3,7
1501 F_{19} F_{79}  yes 19,79 1953 F_{31}  _{2}F_{63} yes  3,7
1561 F_{7} F_{223}  no  1953 _{1}F_{63} _{1}F_{217} yes  3,31
1591 F_{37} F_{43}  no  1953 _{1}F_{63} _{2}F_{217} no
1603 F_{7} F_{229} yes 7,229 1953 _{2}F_{63} _{1}F_{217} no
1629 F_{9} F_{181} no 1953 _{2}F_{63} _{2}F_{217} no
1651 F_{13} F_{127} yes 13,127 1957 F_{19} F_{103}  no
1687 F_{7} F_{241} yes 7,241 1963 F_{13} F_{151}  no
1729 F_{7}  _{1}F_{247} yes  7,19 1981 F_{7} F_{283} yes 7,283
1729 F_{7}  _{2}F_{247} yes  7,19 2007 F_{9} F_{223} yes 3,223
1729 F_{13} _{1}F_{133} yes  7,13 2041 F_{13} F_{157} yes 13,157
1729 F_{13} _{2}F_{133} yes  7,13 2061 F_{9} F_{229} yes 3,229
1729 F_{19}  _{1}F_{91}  no  2077 F_{31} F_{67}  yes 31,67
1729 F_{19}  _{2}F_{91} yes  7,13 2119 F_{13} F_{163}  no

Figure 2 Q-adequacy of Bicyclic Bicubic Fields (cont.)

References

[1] H. Cohen. A Course in Computational Algebraic Number Theory, Springer-Verlag, Berlin, 1993.

[2] A. Kay. Bicyclic bicubic fields, Master’s thesis, Bryn Mawr College, 1994.

[3] N. Koblitz. p-adic Numbers, p-adic Analysis, and Zeta-Functions,2nd ed., Springer, New York, 1984.

[4] D. B. Leep, T. L. Smith, and R. Solomon. Adequate extensions of global fields, preprint.

[5] C. Parry. Bicyclic bicubic fields, Canad. J. Math. 42 (1990), 491–507.

[6] M. Schacher. Subfields of division rings, I, J. Algebra 9 (1968), 451–477.