Galois cohomology for quadratic extensions

Question

Suppose that $L$ is a quadratic extension of $\mathbb{Q}$. Since $L/\mathbb{Q}$ is a Galois extension, a general form of Hilbert Theorem 90 implies that for every $m\geq 1$ $$ H^{1}(\operatorname{Gal}(L/\mathbb{Q}), \operatorname{GL}_{m}(L)) = 1 $$ is the trivial group. In particular, when $m=1$, we get that $$ H^{1}(\operatorname{Gal}(L/\mathbb{Q}), L^{\times}) = 1 $$ What about higher $H^{i}$? Do we know that $$ H^{2}(\operatorname{Gal}(L/\mathbb{Q}), L^{\times}) $$ is also the trivial group? If not, how would one go about computing this group?

Answer 1

For any Galois extension L/K (finite or not), $H^1 (L/K, L^*)$ is trivial. This is the coh. version of Hilbert's theorem 90. If $G$ is a finite cyclic group, say generated by $s$, an abstract periodicity result states that, for any $G$-module $M$, for any $i\in \mathbf Z, H^i(G, M)\cong H^{i-2} (G, M)$, provided we use the Tate modified cohomology groups for $i \le 0$ . For the definition of these, see e.g. Cassels-Fröhlich, chap.IV. Suffice it to say that here, $H^1 (L/K, L^*)\cong Ker N/(L^*)^{(s-1)}$ (thus we recover the classical formulation of Hilbert 90) and $H^2 (L/K, L^*)\cong K^*/N(L^*)$ , where $N$ is the norm map.

It remains to consider specifically extensions of number fields (or more generally global fields). The goal of CFT is to describe the Galois group of an abelian extension $L/K$ in terms of the arithmetic of the base field alone. The central coh. result is : for any Galois extension $L/K$ of degree $n$, there is a canonical isomorphism $H^2 (L/K, C_L)\cong \frac 1n \mathbf Z /\mathbf Z$, where $C_L$ is the so called idèle class group; if $L/K$ is abelian (not necessarily cyclic), there is a canonical isomorphism ("reciprocity isomorphism") $C_K / N(C_L) \cong Gal(L/K)$ (op. cit., chap.VII).

To go further, one must appeal to the global-local principle, i.e. to derive the global reciprocity isomorphism from the local ones, which are more "visible". "Local" here means that one introduces the completions $K_v$ of $K$ wrt. all the places $v$ of $K$, finite or infinite : if $v$ is real (resp. complex) archimedean , $K_v = \mathbf R$ (resp. $\mathbf C$); if $v$ is a place above a prime $p$, $K_v$ is a finite extension of the $p$-adic field $\mathbf Q_p$. Then local CFT (op. cit., chap. VI) asserts the existence of local canonical isomorphisms completely analogous to the global ones, just replacing the idèle class group $C_L$ by the multiplicative group $(L_w) ^{*}$. One can derive from this a description of the natural restriction map $ f_i :H^i (L/K, L^*) \to \oplus H^i(L_w/K_v, (L_w)^*)$ (op. cit., chap. VII, §11.4), and sometimes $H^i (L/K, L^*)$ itself. One particular case is the Hasse normic principle : if $L/K$ is cyclic, then $f_2$ is injective and coker $f_2$ is dual to the kernel of the restriction map $H^1(L/K, \mathbf Z) \to \oplus H^1(L_w/K_v, \mathbf Z)$. These $H^1$ are $Hom$ groups and since the Galois groups are finite, they are null. In the particular case of quadratic fields, everything remaining is explicit. I leave it to you.