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Galois module structure of ideals in wildly ramified cyclic extensions of degree p 2

Gove Griffith Elder (1995)

Annales de l'institut Fourier

For L / K , any totally ramified cyclic extension of degree p 2 of local fields which are finite extensions of the field of p -adic numbers, we describe the p [ Gal ( L / K ) ] -module structure of each fractional ideal of L explicitly in terms of the 4 p + 1 indecomposable p [ Gal ( L / K ) ] -modules classified by Heller and Reiner. The exponents are determined only by the invariants of ramification.

Galois structure of ideals in wildly ramified abelian p -extensions of a p -adic field, and some applications

Nigel P. Byott (1997)

Journal de théorie des nombres de Bordeaux

Let K be a finite extension of p with ramification index e , and let L / K be a finite abelian p -extension with Galois group Γ and ramification index p n . We give a criterion in terms of the ramification numbers t i for a fractional ideal 𝔓 h of the valuation ring S of L not to be free over its associated order 𝔄 ( K Γ ; 𝔓 h ) . In particular, if t n - [ t n / p ] < p n - 1 e then the inverse different can be free over its associated order only when t i - 1 (mod p n ) for all i . We give three consequences of this. Firstly, if 𝔄 ( K Γ ; S ) is a Hopf order and S is 𝔄 ( K Γ ; S ) -Galois...

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