QUATERNION FIELDS of DEGREE 8: Galois Module Structure.

Jacques Martinet Last update: Nov. 24th, 2014; first on line: January 16th, 2015.

Abstract. The note [8] of my homepage, as most of the Notes aux Comptes Rendus de l’Académie des Sciences de Paris, contains almost no proof. I give here detailed proofs for most of the results of [8]. In particular, we shall see that case C in Proposition 1 does not exist, as announced in the preceding Corrigendum.

Some of the proofs could have been found in Fröhlich’s Inventiones paper [Fr], but Fröhlich’s proofs make use of specific techniques from papers he wrote in the fifties, scarcely used by other mathematicians, so that it seems reasonable to write self-contained proofs.

We shall consider as in [8] a quaternionic extension of N/ℚ of degree 8. with quadratic subfields k_{1}, k_{2}, k_{3} and biquadratic subfield K; the notation d_{i}, m_{i}, … is as in Corrigendum, and focus on ramification properties of primes p of ℚ (or p = ∞), with special emphasis on p = 2. However we first consider more generally Galois extensions of a field K_{0} of characteristic κ ≠ 2, with Galois group a subgroup of H_{8} =: ⟨σ, τ | σ^{4} = 1, τ^{2} = σ^{2}, τστ^{−1} = σ^{−1}⟩, thus H_{8} itself or a cyclic group of order 4, 2, or 1 (these subgroups naturally occur when we replace K_{0} by its completion at some place).

1. Embedding problems for H H H_{8 8 8}: general results. By a theorem of Witt ([Wi]), K embeds in a H_{8}-extension N of K_{0} if and only if the quadratic form Q := m_{1}X^{2} + m_{2}Y^{2} + m_{3}Z^{2} is equivalent over K_{0} to Q_{0} := X^{2} + Y^{2} + Z^{2}.

To a quadratic forms ∑_{i=1}^{r} a_{i}X_{i}^{2} on K_{0} one can attach (cohomological) invariants w_{i}, i ≥ 0, of which we only need w_{0} (the rank r ∈ ℤ), w_{1} (the discriminant d = ∏_{i} a_{i} ∈ K_{0}/K_{0}^{×}^{2}), and w_{2} (the Hasse-Witt invariant = ∏_{i<j}(a_{i}, a_{j}) ∈ Br_{2}(K_{0})). [Br is the Brauer group, and the symbol (a, b) is the image (of order 2) of the quaternion algebra over K_{0} defined by i^{2} = a, j^{2} = b, ji = −ij.]

The forms Q and Q_{0} both have rank 3 and discriminant 1 ∈ K_{0}/K_{0}^{×}^{2}. If K_{0} is a local field, a quadratic form is well-defined up to equivalence by w_{0}, w_{1}, w_{2}, and w_{2} is the Hilbert symbol thanks to the identification Br_{2}(K_{0}) = {±1}; and if K_{0} is a number field, then the Hasse-Minkowski theorem shows that equivalence can be read on the completions of K_{0}.

The situation is somewhat similar for cyclic groups of order 4: it is well known (and easy to prove) that a quadratic extension K_{0}(\sqrt{m}) embeds in a cyclic extension of degree 4 if and only if m is the sum of two squares in K_{0}, i.e. if and only if mX^{2} − Y^{2} − Z^{2} represents zero, and in the local case, this is equivalent to (m, −1) = +1.

The following proposition makes easy the calculation of w_{2} for H_{8}- extensions. Proposition We have w_{2}(Q) = (−1, −1)(−m_{1}, −m_{2}). Proof. Since m_{3} ≡m_{1}m_{2} mod K_{0}^{2}, we have (m_{1}, m_{3}) = (m_{1}, m_{1}m_{2}) and (m_{2}, m_{3}) = (m_{2}, m_{1}m_{2}), hence (m_{1}, m_{3})(m_{2}, m_{3}) = (m_{1}m_{2}, m_{1}m_{2}), and by the rule (a, −a) = 1, this is equal to

(m_{1}m_{2}, −1) = (m_{1}, −1)(m_{2}, −1) .

We then have w_{2}(Q) = [(m_{1}, m_{2})(−1, m_{2})(−m_{1}, −1)](−1, −1), and the result follows by bilinearity. □

Over ℝ, we have

(a, b) = −1 if a and b are negative, (a, b) = +1 otherwise. (∗_{∞}) Over ℚ_{p}, the symbol (a, b) can be the calculated using the formulae

below, taken from [Se2]. Write

a = p^{α}u and b = p^{β}v

where u, v are p-units, and for u ∈ ℤ_{2} or ℤ, set

ε(u) = \frac{u−1}{2} mod 2 and ω(u) = \frac{u^{2}−1}{8} mod 2 .

Then we have

p odd : (a, b)_{p} = (−1)^{αβε(p)} (\frac{v}{p})^{α}(\frac{u}{p})^{β}; (∗_{p})
p = 2 : (a, b)_{2} = (−1)^{ε(u)ε(v)+αω(v)+βω(u)} . (∗_{2})

In particular, we have (−1, −1)=−1 over ℝ and ℚ_{2}, and (−1, −1)=+1 otherwise.

[More generally, over an extension L of ℚ_{p}, we have (−1, −1) = −1 if and only if p = 2 and [L : ℚ_{p}] is odd.]

2. Embedding problems for H H H_{8 8 8}: results over Q Q Q and Q Q Q_{p p p}. As a set of representatives for ℚ_{2}^{∗}/ℚ_{2}^{∗}^{2}, we may take

{±1, ±3, ±2, ±6},

defining seven quadratic extensions: ℚ_{2}(\sqrt{−3}), unramified; ℚ_{2}(\sqrt{3}) and ℚ_{2}(\sqrt{−1}), with ramification jump 1; and ℚ_{2}(\sqrt{±2}) and ℚ_{2}(\sqrt{±6}), with ramification jump 2.

[For m_{1} ≡ m_{2} ≡ 1 mod 2 (resp. ≡ 2 mod 4), ℚ_{2}(\sqrt{m_{1}}) ≃ ℚ_{2}(\sqrt{m_{2}}) is equivalent to m_{1} ≡ m_{2} mod 8 (resp. m_{1} ≡ m_{2} mod 16).]

3

The calculation of the symbols (m, −1)_{2} shows that ℚ_{2}(\sqrt{−3}) = ℚ_{2}(\sqrt{5}), ℚ_{2}(\sqrt{2}) and ℚ_{2}(\sqrt{−6}) = ℚ_{2}(\sqrt{10}) embed in cyclic extensions of degree 4 and that the the remaining four do not.

From the list of quadratic extensions we deduce that there are also seven biquadratic extensions of ℚ_{2}, that we list below according to the ramification index e_{2} of 2, giving the corresponding sets {m_{1}, m_{2}, m_{3}}: e_{2} = 2 : {−3, −1, 3}, {−3, 2, −6}, {−3, −2, 6}; e_{2} = 4 : {3, 2, 6}, {3, −2, −6}, {−1, 2, −2}, {−1, 6, −6}.

To list those of these fields which embed in a H_{8}-extension, we may discard from the list above those for which Q represents −1, since Q_{0} does not. We are then left with three biquadratic extensions of ℚ_{2} out of seven, those with

{m_{1}, m_{2}, m_{3}} = {−3, −2, 6} (e_{2} = 2), {3, 2, 6} or {3, −2, −6} (e_{2} = 4) ,

and using (∗_{2}) we check that these extensions do embed in a H_{8} one.

It is a well-known fact (immediate from the very definition of ramification groups) that in a cyclic extension of prime power degree, an ideal which ramifies at some step then ramifies at all the further steps. Putting together this remark and the calculations above, we obtain the following statement, in which n_{2} stands for the local degree and e_{2} (as above) for the ramification index of 2 of the biquadratic subfield of an H_{8}-extension of ℚ:

Theorem. Assume that 2 is ramified in K/ℚ. Then one of the following holds for the completion N̂_{2} of N/ℚ at a prime above 2 of N:

(1) (n_{2}, e_{2}) = (2, 2). N̂_{2}/ℚ_{2} is cyclic of degree 4 and contains ℚ_{2}(\sqrt{2}) or ℚ_{2}(\sqrt{−6}).

(2) (n_{2}, e_{2}) = (4, 2). N̂_{2} contains ℚ_{2}(\sqrt{3}, \sqrt{2}).

(3) (n_{2}, e_{2}) = (4, 4). N̂_{2} contains one of the two fields ℚ_{2}(\sqrt{3}, \sqrt{2}) or ℚ_{2}(\sqrt{3}, \sqrt{±2}).

For the sake of completeness, I give below the embedding conditions at an odd prime p, using {1, u, p, pu} as a set of representatives for ℚ_{p}/ℚ_{p}^{2}, where u ∈ ℤ_{p}^{×} is a non-residue modulo p.

(a) We have (u, −1)_{p} = +1 and (p, −1)_{p} = (pu, −1)_{p} = (\frac{−1}{p}), so that embedding in a cyclic extension of degree 4 is possible for all quadratic extensions if p ≡ 1 mod 4, but uniquely for the unramified one if p ≡ −1 mod 4.

(b) There is a unique biquadratic extension, associated with the set \{u, p, pu\}. Since (u, p)_{p} = (\frac{u}{p}) = −1, this does not embed in a quaternionic extension.

(c) Putting together (a) and (b), we see that an odd prime p which ramifies in K/ℚ ramifies in two quadratic sub-extensions of K/ℚ and splits in the third one.

3. Ramification and discriminants. From now on N stands for an H_{8}-extension of ℚ with quadratic subfields k_{i} = \sqrt{m_{i}}, where the m_{i} are square-free integers, and biquadratic subfield K. Let d_{i} be the discriminant of k_{i} (d_{i} = m_{i} or 4m_{i}) and D = d_{1}d_{2}d_{3} that of K (a square). Finally let Δ be the discriminant of N, related to that of K by the standard transitivity relation

Δ = D^{2} N_{K/ℚ}(δ_{N/K}) .

The behaviour in N/ℚ of the real prime of ℚ is clear: N is totally real or totally imaginary, K is totally real in both cases (because σ^{2} = τ^{2} is the only element of order 2 in H_{8}), and Δ is strictly positive.

From now on let p be a finite prime of ℚ, ramified in N.

We first consider the ramification index of p in N/ℚ. If this index in K/ℚ is 1 (resp. 2, resp. 4), it is 1 or 2 (resp. 4, resp. 8) in N/ℚ, and the latter cases may occur only if p = 2. We keep the following convention of [8]:

(1) If p is not totally ramified in K/ℚ, then p is not ramified in k_{1}.

(2) If p = 2 is totally ramified in K/ℚ, then m_{1} ≡ 3 mod 4 and m_{2} ≡ m_{3} ≡ 2 mod 4.

Assume first that p is odd.

By the results proved at the end of the section above, if p is ramified in K/ℚ, it splits in k_{1}/ℚ and we have pℤ_{N} = (𝔓𝔓′)^{4}, and since the ramification is tame, 𝔓 and 𝔓′ have exponent 3 in the different D_{N/ℚ}. Hence v_{p}(Δ) = 6.

If p is ramified only in N/K, then pℤ_{N} is the square of a product of 4 prime ideals of degree 1 or 2 prime ideals of degree 2, having exponent 1 in D_{N/ℚ}. Hence v_{p}(Δ) = 4.

To deal with p = 2 we shall use the following two results, valid in great generality for any Galois extension M/L relatively to a Dedekind domain A with fraction field L and its integral closure B in M (at least when the residue extensions are separable). Let 𝔭 be a prime ideal in A and 𝔓 a prime ideal in B lying above 𝔭, of ramification index e.

(1) We have G_{i}(𝔓) = \{1\} if i > \frac{v_{𝔓}(p)}{p−1} ([Se1], Sec. IV.2, Exer. (3)).

(2) (The Hasse-Arf theorem.) If M/L is Abelian then the ramification jumps in upper numbering are integral. (This is equivalent to

the existence of strong congruences among the jumps in lower numbering; such congruences are explicitly written in [Se1], Section IV.3; note that the weaker result of Prop. 11 in Section IV.2 sometimes suffices.)

Assume finally that p = 2. Let t_{1} or t_{1}, t_{2} or t_{1}, t_{2}, t_{3} be the ramification jumps. Recall that the exponent of a prime ideal 𝔓 in the different of a Galois extension is given by the formula

v_{𝔓}(D) = ∑_{i≥0} (|G_{i}| − 1) ,

and note that we have t_{1} ≥ 1 since our Galois group is a 2-group.

If e_{2} = 1 (i.e., if 2 is unramified in K/ℚ) we have a single jump t_{1} which satisfies 1 ≤ t_{1} ≤ 2, hence t_{1} = 1 or 2, v_{𝔓}(D_{N/ℚ}) = 2 or 3, and v_{2}(Δ) = 8 or 12, cases (A) or (B) of [7].

If e_{2} = 2, then K̂_{𝔓}/ℚ_{2} is the extension by \sqrt{2} or \sqrt{−6} of an unramified extension of ℚ_{2}, so that t_{1} = 2, hence t_{2} = 4 (because t_{2} ≡ t_{1} mod 2 and t_{2} ≤ 4), whence v_{𝔓}(D_{N/ℚ}) = 3 × (4 − 1) + 2 × (2 − 1) = 11, and v_{2}(Δ) = 22, case (D) of [7] (and case (C) of [7] does not exist).

If e_{2} = 4, then 2 is totally ramified in N/Q, we have G^{1} = G_{1} = G, and G_{2} corresponds to ℚ_{2}(\sqrt{3})/ℚ_{2}, which implies t_{1} = 1 hence t_{2} = 3, then t_{3} = 5 or 7, and indeed t_{3} = 7 by Hasse-Arf applied to N/k_{1}, whence

v_{2}(Δ) = v_{𝔓}(D_{N/ℚ}) = 2 × 7 + 2 × 3 + 4 × 1 = 14 + 6 + 4 = 24 ,

case (E) of [7].

Remark. In all cases the upper ramification jumps are integral. This is clear for primes which are not totally ramified (because the inertia groups are cyclic), and the explicit calculation of Herbrand’s function φ ([Se1], Section IV.3) shows that in the totally ramified case, the upper ramification jumps, namely 1, 2, 3, are integral. However this is not general for non-Abelian extensions; see [Se1], Section IV.3, Exer. 3 for a quaternionic example (over ℚ(i)); many examples can be found in [Fo].

4. The associated order. We keep the notation N, K, … of the previous sections, we set by G = Gal(N/ℚ) (G ≃ H_{8}), and denote by 𝔒_{G} the order in ℚ[G] associated to N:

𝔒_{G} = {λ ∈ ℚ[G] | λℤ_{N} ⊂ ℤ_{N}}.

Let g = G/{1, τ^{2}}. Let H be the skew-field of “ordinary” quaternions over ℚ, with basis (1, i, j, k) (i^{2} = j^{2} = −1, ij = −ji = k) we consider the order 𝔒_{0} = ℤ[i, j, k]. Note that 𝔒_{0} is contained in a unique maximal order, namely the Hurwitz order 𝔐 = 𝔒_{0} ∪ 𝔒_{0} + ω, ω = \frac{−1+i+j+k}{2}.

Consider in ℚ[G] the two central idempotents

e′ = \frac{1−σ^{2}}{2} and e′′ = \frac{1+σ^{2}}{2} .

They are orthogonal and add to 1, hence split ℚ[G] as

ℚ[G] = ℚ[G]e′ × ℚ[G]e′′ ≃ ℚ[g] × H.

These idempotents split N into a direct sum N = K ⊥ K′ (orthogonal for the bilinear form trace; K′, not a field, is the kernel of Tr_{N/ℚ}). We now show that when 2 is ramified, these idempotent split ℤ_{N} into the direct sum of ℤ_{K} and the set ℤ_{K′} of integral elements of K′.

Let t be the highest ramification jump of some ideal 𝔓 | 2. It results from Section 3 that G_{t} is the center {1, τ^{2}} of G and that t+1 = e_{𝔓}, the ramification index of 𝔓. The definition of ramification groups shows that we have on ℤ_{N} the congruence τ^{2}x ≡ x mod 𝔓^{e} for every 𝔓, hence τ^{2} ≡ x mod 2, that is, e′ℤ_{N} ⊂ ℤ_{N} ∩ K = ℤ_{K}, and finally e′ℤ_{N} = \frac{1}{2} Tr_{N/K}(ℤ_{N}) = ℤ_{K}.

Thanks to this splitting of ℤ_{N} the projection on ℤ[g] of 𝔒_{G} is the order 𝔒_{g} associated to ℤ_{K} in ℚ[g]; see Corrigendum. The discriminant calculation of the note then shows that 𝔒_{G} = 𝔒_{g} × 𝔒_{0}.

The result above completes the proofs of the note except that of Proposition 2. I leave it to the reader, and state instead its generalization to an arbitrary rank Proposition 2’. (Plesken, [Pl].) Let M be a finitely generated (left, torsion free)) module over 𝔒_{0}.Then M is isomorphic to a direct sum 𝔒_{0}^{p} ⊕ 𝔐^{q} (where r := p + q is the rank of M).

Results of this kind could be considered in the following setting: first establish that projective modules (over a convenient order) are locally free, then calculate the kernel in class groups of extension map to a maximal order.

References

[Fo] J.-M. Fontaine, Groupes de ramification et représentation d’Artin, Ann. Sci. E.N.S. 4 (1971), 337–392.

[Fr] A. Fröhlich, Artin Root Numbers and Normal Integral Bases for Quaternion Fields, Invent. Math 17 (1972), 143–166.

[Pl] W. Plesken, Private communication (Bonn, May 1991).

[Se1] J.-P. Serre, Corps locaux, Hermann, Paris (1962; 4-ième éd. : 2004). English ed.: Local Fields, Springer, Heidelberg, GTM 67 (1979).

[Se2] J.-P. Serre, Cours d’Arithmétique, P.U.F., Paris (1970; 4-ième éd. : 1995). English ed.: A Course in Arithmetic, Springer, Heidelberg, GTM 7 (1996).

[Wi] E. Witt, Konstruktion von galoisschen Körpern der Characteristik p zu vorgegebener Gruppe der Ordnung p^{f}, J. reine angew. math. 176 (1937), 31–44, = Ges. Abh., Springer, Heidelberg (1998), 120–128.