Galois theory in algebras over finite fields - applications to the Berlekamp algebra
Preda Mihăilescu (University Göttingen, Germany)
It is a useful pattern to interpret an Fp-algebra A = Fp[ X ]/(f(X)), with f ∈ Fp[ X ], as the image from some global extension in the following sense: there is an p-unramified global galois extension L/K in which p is totally split, and a prime PK ⊆ K above p, such that:
- L = K[ X ]/(g1), for some polynomial g1 ∈ O(K)[ X ] with g1 mod PK = f.
- A = O(K)[ X ]/ (PK, g1 O(K) )[ X ])
In such a situation the galois group Aut( A/Fp ) is generated by Frobenius on the one side and the reduction of the global galois group G=Gal(L/K), on the other side. Since the latter can be evaluated using some polynomials which can be computed in the complex plane, we have a ressource for relaxing the use of evaluations of the Frobenius. This is interesting for small extension degrees and large p.
There are different applications to this idea. I will talk about a new road - which might be better investigated by May - that yields a deterministic variant of the Berlekamp algorithm in some cases in which the lift L/K is well controlled.
Talk slides (PDF).