The conductor of a cyclic quartic field using Gauss sums

Abstract

Let Q denote the field of rational numbers. Let K be a cyclic quartic extension of Q. It is known that there are unique integers A, B, C, D such that $K = Q\left( {\sqrt {A(D + B\sqrt D )} } \right)$ where A is squarefree and odd, D=B ²+C ² is squarefree, B $>$ 0 , C $>$ 0, GCD(A,D)=1. The conductor f(K) of K is f(K) = 2^l|A|D, where $l = \left\{ \begin{gathered} 3,{\text{ if }}D \equiv 2{\text{ }}({\text{mod 4}}){\text{ or }}D \equiv 1{\text{ (mod 4), }}B \equiv 1{\text{ }}({\text{mod 2}}), \hfill \\ 2,{\text{ if }}D \equiv 1{\text{ (mod 4), }}B \equiv 0{\text{ (mod 2), }}A + B \equiv 3{\text{ (mod 4),}} \hfill \\ 0,{\text{ if }}D \equiv 1{\text{ (mod 4), }}B \equiv 0{\text{ (mod 2), }}A + B \equiv 1{\text{ (mod 4)}}{\text{.}} \hfill \\ \end{gathered} \right.$ A simple proof of this formula for f(K) is given, which uses the basic properties of quartic Gauss sums.