In this short post I wish to record the proof of Hilbert’s Theorem 90 by interpreting it as the 1-dimensional case of Galois descent. The ideas all come from the Arcata’ in SGA 4 1/2. Note that in some sense this completes the Kummer Theory argument in the previous post, except now I am assuming Galois Descent as a black box instead. Eventually, I intend to upload to this blog an article which will include a proof of Galois Descent (modelled on either the proof of Grothendieck’s fpqc descent or the more general monadic descent theory), so then finally everything will be in place :). Alternatively, there are great articles about this elsewhere online. Maybe this is a good example of where you should generalise a problem so much it becomes easy.

Firstly, what is the statement of Galois Descent? Let ${L/k}$ be an algebraic Galois extension, with Galois group ${G}$. Consider an ${L}$-vector space ${V}$. A semilinear ${G}$-action on ${V}$ is a set of ${k}$-endomorphisms ${\{\psi_\sigma\}}$ of ${V}$ indexed by elements of ${G}$ such that

$\displaystyle \psi_\sigma(\lambda v) = \sigma(\lambda) \psi_\sigma(v),$

and respecting the group operation on indices: ${\psi_{\tau \sigma} = \psi_\tau \circ \psi_\sigma}$. The semilinearity’ condition could be viewed as saying that ${\psi_\sigma}$ is an ${L}$-linear map ${L\otimes_L V \rightarrow V}$ where the map used to tensor ${L}$ with itself is ${\sigma: L \rightarrow L}$ (rather than the identity map, which would be more conventional).

Given an ${L}$-vector space ${V}$ equipped with a semilinear ${G}$-action, we can consider the ${k}$-vector space ${V^G = \{v \in V| \psi_\sigma(v) = v \ \forall \sigma \in G\}}$ of elements which are invariant under the semilinear action. Conversely, given a ${k}$-vector space ${W}$ we can tensor up to get an ${L}$ vector space ${W \otimes_k L}$ and this has a natural semilinear ${G}$-action given by letting ${G=Gal(L/k)}$ act in the natural manner on the second factor of the tensor product. If ${W}$ is a ${k}$-subspace of an ${L}$ vector-space ${V}$ and this tensor product corresponds to a natural isomorphism ${W \otimes_k L \rightarrow v}$, this gives us a semilinear ${G}$-action on ${V}$.

Theorem 1 (Galois Descent) Fix an ${L}$-vector space ${V}$ of dimension ${n}$. There is a natural bijection, given by the maps described above, between ${n}$-dimensional ${k}$-subspaces of ${V}$ and semilinear ${G}$-actions of ${L}$.

We will use this to prove Hilbert’s Theorem 90, viewing it as the 1-dimensional case of this theorem. In fact, given Hilbert 90 underpins Kummer Theory which underpins Class Field Theory, perhaps it is worth not-so-wildly speculating that more general Galois descent should be expected to underpin the study of nonabelian Galois representations, and therefore understanding these reductions is possibly important as a model for work on Langlands. Anyway, let us recall its statement.

Theorem 2 (Hilbert 90) Suppose ${\chi: Gal(L/k) \rightarrow L^*}$ is a cocycle: a function such that ${\chi(\tau\sigma) = \chi(\tau) \tau \chi(\sigma)}$. Then it is a coboundary: ${\chi(\sigma) = \sigma(x)/x}$ for some ${x \in L*}$. In terms of Galois cohomology, this can be simply stated as

$\displaystyle H^1(Gal(L/k), L^*) = 0.$

Proof: Consider ${V=L}$ as a 1-dimensional vector space over itself, and pick some distinguished nonzero vector ${v_0}$. A cocycle ${\chi}$ can be interpreted as precisely a semilinear ${G}$-action on ${V}$. Indeed, given ${\chi}$, we set ${\psi_\sigma(v_0) = \chi(\sigma) v_0}$ and extending semilinearly, obtain a semilinear ${G}$-action. Indeed:

$\displaystyle \psi_{\tau \sigma}(\lambda v_0) = \tau\sigma(\lambda) \chi(\tau \sigma) v_0 = \tau\sigma(\lambda)\chi(\tau)\tau(\chi(\sigma)) v_0 = \psi_{\tau}(\sigma(\lambda) \chi(\sigma) v_0) = \psi_{\tau} \circ \psi_\sigma (\lambda v_0).$

Conversely, one may check that a semilinear ${G}$-action gives rise, once we have fixed ${v_0}$, to a coycle.

But Galois descent tells us that each such semilinear group action has a ${1}$-dimensional ${k}$-vector subspace, fixed by the action. Let ${v=\mu v_1}$ be a basis vector for this subspace (so ${\mu \not= 0}$).

Then for all ${\sigma \in G}$,

$\displaystyle \mu v_1 = \psi_\sigma(\mu v_1) = \sigma(\mu) \chi(\sigma) v_1,$

proving that ${\chi}$ is in fact a coboundary, corresponding to ${\mu \in L^*}$. $\Box$

In fact, by doing a similar thing but working in ${n}$ dimensions and picking a larger basis at the start (and then working with matrices appropriately), it looks like we should be able to generalise Hilbert 90 to the following statement.

Proposition 3 (Generalised Hilbert 90?) Let ${L/k}$ be an algebraic Galois extension with Galois group ${G}$, ${n}$ a positive integer. Then

$\displaystyle H^1(G, GL_n(L^*)) = 0.$

Presumably this is well-known, and maybe it gives rise to a natural generalisation of Kummer theory, but this post is now longer than I was planning anyway, so I think I’ll leave it there for now.