Classification of the Hopf Galois Structures on Prime Power Radical Extensions

✍ Scribed by Timothy Kohl

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 264 KB
• Volume: 207
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7479

✦ Synopsis

Let p be an odd prime and n a positive integer and let k be a field of Ž .

r p and let r denote the largest integer between 0 and n such that K l k s p Ž .

r r r k , where denotes a primitive p th root of unity. The extension Krk is p p separable, but not necessarily normal and, by Greither and Pareigis, is H-Galois ˜with H a K-Hopf algebra form of a group ring kN where K is the normal closure Ž . of Krk. H is said to be almost classical if N -Gal Krk . The result is that if rn then there are p r Hopf Galois structures on Krk for which the associated group N is cyclic of order p n . Of these, p m inŽ r, nyr . are almost classical and the rest are non-almost classical. When r s n, there are p ny 1 H-Galois structures for which N ( C n of which only one is almost classical. Finally, we show that these p are the only structures possible. That is, for this class of extensions, N must be cyclic.