4.4 Desarrollo de la Propuesta
4.4.7 El Acuerdo Reparatorio
Definition 6.7(Klein four-group). AKlein four-groupis a group isomorphic to (Z/2Z)⊕(Z/2Z).
Lemma 6.8. LetA be a Klein four-group. The map
{skew abelian groups (A,{±1}, β)} →A
that sends a skew abelian group [Definition 2.7]of the form (A,{±1}, β)to its skew element [Definition 2.12] is a bijection.
Proof. Let (A,{±1}, β) be a skew abelian group and letgbe its skew element. Since the antisymmetric pairingβtakes values in{±1}, giving the annihilator of each element in A is equivalent to giving the map β. For every nonzero element a∈A the annihilatora⊥ ofa has order 2. We havea+g ∈a⊥ and
g∈g⊥. Sincea+g= 0 is equivalent toa=g, we get an explicit description of the annihilator of any element in Athat depends only ong. This shows that the map in the statement of Lemma 6.8 is injective. The classification of skew abelian groups [Theorem 2.28] implies that it is also surjective.
Notation 6.9. LetK be a quadratic number field, let ∆ be its discriminant, letOK be the ring of integers ofK, and letGbe the Galois group Gal(K/Q).
Theorem 6.10. Let the notation be as in Notation 6.9. For each place v of
K above 2 let Kv be the completion of K at v, let Uv be the unit group of
the ring of integers of Kv, and let Uv⊥ be the annihilator in Kv∗ of Uv with
respect to the norm-residue symbol (·,·)Kv,2 : K
∗
v ×Kv∗ → {±1}. Then the
groups(OK/4OK)∗/(OK/4OK)∗
2
andQ
v|2Uv/Uv⊥ are Klein four-groups and
the natural group isomorphism
(OK/4OK)∗/(OK/4OK)∗
2 ∼
−→Y
v|2
Uv/Uv⊥ (6.11)
given by Theorem 5.12is a (Z/2Z)[G]-module isomorphism and is an isomor-
phism of skew abelian groups
((OK/4OK)∗/(OK/4OK)∗ 2 ,{±1}, β)−∼→(Y v|2 Uv/Uv⊥,{±1}, Y v|2 (·,·)Kv,2),
where β is the unique antisymmetric perfect pairing
β: (OK/4OK)∗/(OK/4OK)∗ 2 ×(OK/4OK)∗/(OK/4OK)∗ 2 → {±1} that makes ((OK/4OK)∗/(OK/4OK)∗ 2
,{±1}, β) into a skew abelian group with skew element(−1 + 4OK)·(OK/4OK)∗
2
6.4. The two-adic component of the unit residue group
Proof. Since the group isomorphism given by Theorem 5.12 respects the nat- ural actions ofGon (OK/4OK)∗/(OK/4OK)∗
2
and onQ
v|2Uv/Uv⊥, we get a natural (Z/2Z)[G]-module isomorphism
(OK/4OK)∗/(OK/4OK)∗
2 ∼
−→Y
v|2
Uv/Uv⊥.
Corollary 3.91 gives a group isomorphismQ
v|2Uv/Uv⊥∼= (Z/2Z)2, because the sum of the local degrees P
v|2[Kv :Q2], where Q2 is the field of 2-adic ratio-
nals, equals the global degree [K:Q]. SinceQv|2(·,·)Kv,2is an antisymmetric
perfect pairing, there is an antisymmetric perfect pairing
β: (OK/4OK)∗/(OK/4OK)∗
2
×(OK/4OK)∗/(OK/4OK)∗
2
→ {±1}
such that the group isomorphism 6.11 is an isomorphism of skew abelian groups ((OK/4OK)∗/(OK/4OK)∗2,{±1}, β) ∼ −→(Y v|2 Uv/Uv⊥,{±1}, Y v|2 (·,·)Kv,2).
Letβbe such an antisymmetric perfect pairing. Corollary 3.92 and Lemma 6.8 imply that β is the unique antisymmetric perfect pairing
β: (OK/4OK)∗/(OK/4OK)∗ 2 ×(OK/4OK)∗/(OK/4OK)∗ 2 → {±1} that makes ((OK/4OK)∗/(OK/4OK)∗ 2
,{±1}, β) into a skew abelian group with skew element (−1 + 4OK)·(OK/4OK)∗
2
.
Theorem 6.12. Let the notation be as in Notation 6.9. Then one has the following.
(a) For∆≡1 mod 4 there is a(Z/2Z)[G]-module isomorphism
(OK/4OK)∗/(OK/4OK)∗
2 ∼
−→ OK/2OK
of free(Z/2Z)[G]-modules of rank 1such that for eacha∈ OK one has (1 + 2a+ 4OK)·(OK/4OK)∗
2
7→a+ 2OK. (b) For∆≡0 mod 8there is a (Z/2Z)[G]-module isomorphism
(Z/2Z)[G]−∼→(OK/4OK)∗/(OK/4OK)∗2
of free (Z/2Z)[G]-modules of rank1 such that for each pair(a0, a1)∈(Z/2Z)2
one has
a0+a1σ7→(1 +p∆/4 + 4OK)a0(1−p∆/4 + 4OK)a1·(O
K/4OK)∗2,
Chapter 6. Quadratic number fields and a biquadratic example
(c) For∆≡4 mod 8there is a(Z/2Z)[G]-module isomorphism
(OK/4OK)∗/(OK/4OK)∗
2 ∼
−→(OK/2OK)∗× h5 mod 8i
of(Z/2Z)[G]-modules with trivialG-actions such that for eacha∈ OKcoprime
to2 one has
(a+ 4OK)·(OK/4OK)∗
2
7→(a+ 2OK,NK/Qamod 8),
where NK/Q:K→Qdenotes the norm map from K toQ.
Proof. We prove each case separately.
(a) Since the rational prime 2 does not divide ∆, the extension K/Q is
unramified at 2. The result follows from Lemma 6.6.
(b) By Theorem 6.10 the group (OK/4OK)∗/(OK/4OK)∗
2 is a Klein four- group. We have NK/Q(1 + p ∆/4) = NK/Q(1− p ∆/4) = 1−∆/4≡ −1 mod 4.
Since by direct computation −1 + 4OK is not a square in (OK/4OK)∗, the residue classes of 1 +p∆/4 and 1−p
∆/4 in (OK/4OK)∗/(OK/4OK)∗
2
are different. The nontrivial action of σ on the set {1 +p∆/4,1−p
∆/4}
implies that these residue classes are both different from the residue class of 1. The (Z/2Z)[G]-module isomorphism in the statement of (b) in Theorem 6.12 follows.
(c) Letσbe the generator ofG. Since for everyb∈ OK we have NK/Q(1 + 4b) = 1 + 4(b+σ(b)) + 16bσ(b)≡1 mod 8
and for eacha∈ OK coprime to 2 we have
NK/Q(a2) = (aσ(a))2≡1 mod 8,
the map in the statement of (c) in Theorem 6.12 is well-defined. It is surjective, because it is a group homomorphism, it is surjective on the first component, and we have
NK/Q(1 + 2
p
∆/4) = 1−∆≡5 mod 8.
The injectivity follows from the fact that by Theorem 6.10 it is a surjective group homomorphism between two finite groups of the same order. This map is a (Z/2Z)[G]-module isomorphism of (Z/2Z)[G]-modules with trivialG-actions,
because it respects the natural G-actions on both groups and the natural