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 be an algebraic Galois extension, with Galois group . Consider an -vector space . A *semilinear* -action on is a set of -endomorphisms of indexed by elements of such that

and respecting the group operation on indices: . The `semilinearity’ condition could be viewed as saying that is an -linear map where the map used to tensor with itself is (rather than the identity map, which would be more conventional).

Given an -vector space equipped with a semilinear -action, we can consider the -vector space of elements which are invariant under the semilinear action. Conversely, given a -vector space we can tensor up to get an vector space and this has a natural semilinear -action given by letting act in the natural manner on the second factor of the tensor product. If is a -subspace of an vector-space and this tensor product corresponds to a natural isomorphism , this gives us a semilinear -action on .

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

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 is a cocycle: a function such that . Then it is a coboundary: for some . In terms of Galois cohomology, this can be simply stated as

*Proof:* Consider as a 1-dimensional vector space over itself, and pick some distinguished nonzero vector . A cocycle can be interpreted as precisely a semilinear -action on . Indeed, given , we set and extending semilinearly, obtain a semilinear -action. Indeed:

Conversely, one may check that a semilinear -action gives rise, once we have fixed , to a coycle.

But Galois descent tells us that each such semilinear group action has a -dimensional -vector subspace, fixed by the action. Let be a basis vector for this subspace (so ).

Then for all ,

proving that is in fact a coboundary, corresponding to .

In fact, by doing a similar thing but working in 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 be an algebraic Galois extension with Galois group , a positive integer. Then

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.

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January 2, 2012 at 5:42 pm

Martin OrrYour Proposition 3 is well-known – I think it is in Serre’s ‘Local Fields’.