The ring of integers of Hopf-Galois degree p extensions of p-adic fields with dihedral normal closure

Abstract: For an odd prime number

p

, we consider degree

p

extensions

L / K

of

p

-adic fields with normal closure

w i d e t i l d e L

such that the Galois group of

w i d e t i l d e L / K

is the dihedral group of order

2 p

. We shall prove a complete characterization of the freeness of the ring of integers

m a t h c a l O_{L}

over its associated order

m a t h f r a k A_{L / K}

in the unique Hopf-Galois structure on

L / K

, which is analogous to the one already known for cyclic degree

p

extensions of

p

-adic fields. We shall derive positive and negative results on criteria for the freeness of

m a t h c a l O_{L}

as

m a t h f r a k A_{L / K}

-module.

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