The Hopf Galois Property in Subfield Lattices

Abstract

Let K/k be a finite separable extension, n its degree and its Galois closure. For n ≤ 5, Greither and Pareigis show that all Hopf Galois extensions are either Galois or almost classically Galois and they determine the Hopf Galois character of K/k according to the Galois group (or the degree) of . In this paper we study the case n = 6, and intermediate extensions F/k such that , for degrees n = 4, 5, 6. We present an example of a non almost classically Galois Hopf Galois extension of ℚ of the smallest possible degree and new examples of Hopf Galois extensions. In the last section we prove a transitivity property of the Hopf Galois condition.

Key Words:

• Holomorph
• Hopf algebra
• Hopf Galois extension

2010 Mathematics Subject Classification:

• 16T05
• 12F10
• 11R32

View correction statement:

Corrigendum

ACKNOWLEDGMENTS

We thank the referee for a careful reading of our manuscript and valuable comments on it.

Notes

Communicated by V. A. Artamonov.