Repka, Joe Quadratic subfields of quartic extensions of local fields. (English) Zbl 0635.12008 Int. J. Math. Math. Sci. 11, No. 1, 1-4 (1988). 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 Keywords:local class field theory; extension of local fields of degree 4; splitting field of an irreducible equation PDFBibTeX XMLCite \textit{J. Repka}, Int. J. Math. Math. Sci. 11, No. 1, 1--4 (1988; Zbl 0635.12008) Full Text: DOI EuDML