A Note on the Construction of Metacyclic Extensions

Abstract

Let $p$ be an odd prime and $r$ a divisor of $p-1$. We present a characterization of metacyclic extensions of degree $pr$ containing a given cyclic extension of degree $r$ over a field of characteristic other than $p$. Furthermore, we give a method of constructing polynomials with Galois groups which are Frobenius groups of degree $p$.