Purely inseparable extensions of unique factorization domains

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 ...

Unique Factorization Monoids and Domains

It is the purpose of this paper to construct unique factorization (uf) monoids and domains. The principal results are: (1) The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. (2) If M is an ordered monoid and F is a field, the ring ^[[iW"]] of all formal power series with well-ordered support is a uf-domain iff M is naturally ordered (i.e...

Lectures On Unique Factorization Domains

Unique Factorization in Integral Domains

Throughout R is an integral domain unless otherwise specified. Let A and B be sets. We use the notation A ⊆ B to indicate that A is a subset of B and we use the notation A ⊂ B to mean that A is a proper subset of B. The group of elements in R which have a multiplicative inverse (the group of units of R) is denoted R×. Since R has no zero divisors cancellation holds. If a, b, c ∈ R and a 6= 0 th...

Scaffolds and integral Hopf Galois module structure on purely inseparable extensions

Let p be prime. Let L/K be a finite, totally ramified, purely inseparable extension of local fields, [L : K] = p, n ≥ 2. It is known that L/K is Hopf Galois for numerous Hopf algebras H, each of which can act on the extension in numerous ways. For a certain collection of such H we construct “Hopf Galois scaffolds” which allow us to obtain a Hopf analogue to the Normal Basis Theorem for L/K. The...