Details

Cornelius Greither ^[a]

A note on normal bases for the square root of the codifferent in local fields

Pages: 87–109
Received: 12 April 2021
Accepted in revised form: 14 September 2021
Mathematics Subject Classification: 11R33, 11S23.
Keywords: Normal integral bases, inverse different, elliptic functions.
Authors address:
[a]: Fakultät Informatik, Universität der Bundeswehr München, 85577 Neubiberg, Germany.

Abstract: We consider \(p\)-elementary weakly ramified extensions \(M/K\) of \(p\)-adic fields. (Weak ramification is usually wild, but in some sense "not too wild".) It has been known that for odd primes \(p\), the inverse square root \(A_{M/K}\) of the different of \(M/K\) has a normal integral basis, which opens up a nice parallel to the classical theory of tame extensions. In this note, we extend results of Pickett, concerning the existence of normal integral bases for \(A\) with extra qualities. We assume \(K\) to be unramified over \(\mathbb Q_p\) and construct a self-dual normal basis in the case \(M/K\) is cyclic (Pickett treated the case \(K=\mathbb Q_p\)). We give an explicit construction of a normal basis, presumably no longer self-dual, when \(M/K\) is general \(p\)-elementary. The note ends with a few comments concerning the global situation, which is far less understood.

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