Wilson's theorem for finite fields

The Freiman-Ruzsa Theorem in Finite Fields

Let G be a finite abelian group of torsion r and let A be a subset of G. The Freiman-Ruzsa theorem asserts that if |A + A| ≤ K|A| then A is contained in a coset of a subgroup of G of size at most Kr 4 |A|. It was conjectured by Ruzsa that the subgroup size can be reduced to r for some absolute constant C ≥ 2. This conjecture was verified for r = 2 in a sequence of recent works, which have, in f...

Tate’s Isogeny Theorem for Abelian Varieties over Finite Fields

A Structure Theorem for Plesken Lie Algebras over Finite Fields

W. Plesken found a simple but interesting construction of a Lie algebra from a finite group. Cohen and Taylor posed themselves the question of what the Plesken Lie algebra, which is the Lie subalgebra of the group algebra k[G] generated by the elements g − g−1, could be. The result is very fascinating: It turns out that the Lie algebra decomposition of the Plesken Lie algebra into simple Lie al...

Generalizations of the Normal Basis Theorem of Finite Fields

We present a combinatorial characterization of sets of integers (rO,rl, ... ,TII-d, with OSri~"_2, " such that a'O,({1, ... ,a'·-1 fonn a basis of GF(q") over GF(q) for some ae GF'(q"). We use this characterization to prove the following generalization of the nonnal basis theorem fpr finite -fields of characteristic two: Let Ao.Al; ... ,A.,.-1 be integers in the range~<q, with at most one A.; e...

Differential Puiseux theorem in generalized series fields of finite rank

We study differential equations F (y, . . . , y) = 0 where F (Y0, . . . , Yn) is a formal series in Y0, . . . , Yn with coefficients in some field of generalized power series Kr with finite rank r ∈ N ∗. Our purpose is to understand the connection between the set of exponents of the coefficients of the equation Supp F and the set Supp y0 of exponents of the elements y0 ∈ Kr that are solutions.