Hopf-Galois module structure of tamely ramified radical extensions of prime degree

2018/12/21 by Truman, Paul J
#11R33 #16S40 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1812.09394

Abstract

Let K be a number field and let L/K be a tamely ramified radical extension of prime degree p . If K contains a primitive p^th root of unity then L/K is a cyclic Kummer extension; in this case the group algebra K[G] (with G=Gal(L/K) ) gives the unique Hopf-Galois structure on L/K , the ring of algebraic integers \mathfrakO_L is locally free over \mathfrakO_K[G] by Noether's theorem, and Gómez Ayala has determined a criterion for \mathfrakO_L to be a free \mathfrakO_K[G] -module. If K does not contain a primitive p^th root of unity then L/K is a separable, but non-normal, extension, which again admits a unique Hopf-Galois structure. Under the assumption that p is unramified in K , we show that \mathfrakO_L is locally free over its associated order in this Hopf-Galois structure and determine a criterion for it to be free. We find that the conditions that appear in this criterion are identical to those appearing in Gómez Ayala's criterion for the normal case.

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