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.