Invariance in inseparable Galois theory

Hopf Galois structures on primitive purely inseparable extensions

Let L/K be a primitive purely inseparable extension of fields of characteristic p, [L : K] > p, p odd. It is well known that L/K is Hopf Galois for some Hopf algebra H, and it is suspected that L/K is Hopf Galois for numerous choices of H. We construct a family of K-Hopf algebras H for which L is an H-Galois object. For some choices of K we will exhibit an infinite number of such H. We provide ...

Galois invariance, trace codes and subfield subcodes

Article history: Received 26 February 2009 Revised 21 September 2009 Available online 2 February 2010 Communicated by W. Cary Huffman

Galois Invariance of Local Root Numbers

It is a standard remark that the group Aut(C) of abstract field automorphisms of C has uncountably many elements but only two that are are continuous. One consequence is that the analytic properties of a Dirichlet series ∑ n>1 a(n)n −s do not usually carry over to ∑ n>1 ι(a(n))n −s for ι ∈ Aut(C). Silly counterexamples abound: for instance, take a(n) = (1 − √ 2) and choose ι so that ι( √ 2) = −...

Galois Theory

Galois Theory

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set...