STNB2026 (39th edition)
Hopf-Galois structures on separable extensions
Presenters
Cornelius Greither
Abstract
The second lecture of C.G.~builds on the concepts introduced in the first; its essential topics are the so-called Greither-Pareigis correspondence, and the Byott translation. The former assumes that the $K$ algebra $A=L$ is a field, and describes all Hopf Galois situations involving $L$ in terms of $G$ (the Galois group of the normal closure of $L$), $G'$ (the subgroup fixing $L$), and the group of permutations of the $G$-set $G/G'$. This is nice in principle and totally explicit, but it involves groups which are often unwieldy because of their size. Byott's translation reformulates this correspondence in a different and potentially more manageable way, which also involves groups and actions, a key notion being the holomorph of a finite group. We will try to give typical and meaningful examples all along. In particular we will see that for example, field extensions obtained by adjoining cube roots are typically not classically Galois, but can be endowed with a nice Hopf Galois structure. This example belongs to the larger class of so-called ``almost classically Galois extensions''.; those will be treated more generally if time permits. This lecture will be followed by a final lecture, given by Daniel Gil Mu\~noz.
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