Towards a Generalisation of Noether's Theorem to Nonclassical Hopf-Galois Structures

Details Towards a Generalisation of Noether's Theorem to Nonclassical Hopf-Galois Structures Towards a Generalisation of Noether's Theorem to Nonclassical Hopf-Galois Structures

2010-01-11

arxiv.org/abs/1001.1639v1

New York J. Math 17 (2011), 799-810

Mathematics

Number Theory

15 Pages

Scientific paper

We study the nonclassical Hopf-Galois module structure of rings of algebraic integers in some extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is an unramified extension of $ p $-adic fields which is $ H $-Galois for some Hopf algebra $ H $ then $ \OL $ is free over its associated order $ \AH $ in $ H $. If $ H $ is commutative, we show that this conclusion remains valid in ramified extensions of $ p $-adic fields if $ p $ does not divide the degree of the extension. By combining these results we prove a generalisation of Noether's theorem to nonclassical Hopf-Galois structures on domestic extensions of number fields.

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