14,704,208 research outputs found
The valuation criterion for normal basis generators
If is a finite Galois extension of local fields, we say that the
valuation criterion holds if there is an integer such that every
element with valuation generates a normal basis for .
Answering a question of Byott and Elder, we first prove that holds if
and only if the tamely ramified part of the extension is trivial and
every non-zero -submodule of contains a unit. Moreover, the integer
can take one value modulo only, namely , where
is the valuation of the different of . When has positive
characteristic, we thus recover a recent result of Elder and Thomas, proving
that is valid for all extensions in this context. When
\char{\;K}=0, we identify all abelian extensions for which is
true, using algebraic arguments. These extensions are determined by the
behaviour of their cyclic Kummer subextensions
- β¦