×

Quadratic subfields of quartic extensions of local fields. (English) Zbl 0635.12008

By applying some basic facts of local class field theory it is shown that if \(E/F\) is an extension of local fields of degree 4, then there exists a proper intermediate field provided that the residue characteristic is odd. As a consequence one gets that neither the \(A_ 4\) nor the \(S_ 4\) are realizable as Galois groups over F, so, a fortiori, the splitting field of an irreducible equation over F has degree 4 or 8. Counterexamples are given for the residue characteristic 2.
Reviewer: J.Ritter

MSC:

11S15 Ramification and extension theory
11S31 Class field theory; \(p\)-adic formal groups
12E12 Equations in general fields
PDFBibTeX XMLCite
Full Text: DOI EuDML