On, around, and beyond Frobenius' theorem on division algebras

A Frobenius-Schur theorem for Hopf algebras

In this note we prove a generalization of the Frobenius-Schur theorem for finite groups for the case of semisimple Hopf algebra over an algebraically closed field of characteristic 0. A similar result holds in characteristic p > 2 if the Hopf algebra is also cosemisimple. In fact we show a more general version for any finite-dimensional semisimple algebra with an involution; this more general r...

Bilinear Forms on Frobenius Algebras

We analyze the homothety types of associative bilinear forms that can occur on a Hopf algebra or on a local Frobenius k-algebra R with residue field k. If R is symmetric, then there exists a unique form on R up to homothety iff R is commutative. If R is Frobenius, then we introduce a norm based on the Nakayama automorphism of R. We show that if two forms on R are homothetic, then the norm of th...

Frobenius kernel and Wedderburn's little theorem

We give a new proof of the well known Wedderburn's little theorem (1905) that a finite‎ ‎division ring is commutative‎. ‎We apply the concept of Frobenius kernel in Frobenius representation theorem in finite group‎ ‎theory to build a proof‎.

On Nonassociative Division Algebras^)

On normalizers of maximal subfields of division algebras

‎Here‎, ‎we investigate a conjecture posed by Amiri and Ariannejad claiming‎ ‎that if every maximal subfield of a division ring $D$ has trivial normalizer‎, ‎then $D$ is commutative‎. ‎Using Amitsur classification of‎ ‎finite subgroups of division rings‎, ‎it is essentially shown that if‎ ‎$D$ is finite dimensional over its center then it contains a maximal‎ ‎subfield with non-trivial normalize...