Some contributions to the theory of cyclic quartic extensions of the rationals

K is a cyclic quartic extension of Q iff $\text{K = Q((rd + p d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>)</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where d > 1, p and r are rational integers, d squarefree, for which p² + q² = r²d for some integer q. Following a paper of A. A. Albert we show that the absolute discriminant, $\text{d(}\text{K}\text{Q}\text{)}$, of the general cyclic quartic extension is given by $\text{d(}\text{K}\text{Q}\text{) = (W}{}^2\text{d}{}^2\text{)}$ for an explicitly computable rational integer W. We next find that the relative discriminant, $\text{d(}\text{K}\text{F}\text{)}$, is given by $\text{d(}\text{K}\text{F}\text{) = (W d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where $\text{F = Q (d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ is K′s uniquely determined quadratic subfield. We use this last result in conjunction with Corollary 3, page 359, of Narkiewicz's “Elementary and Analytic Theory of Algebraic Numbers” (PWN-Polish Scientific Publishers, 1974) to establish the following Theorem 1: If the (wide) class number of$\text{F = Q(d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$is odd then every cyclic quartic extensionKofQcontainingFhas a relative integral basis overF. We give a second, more organic, proof of Theorem 1 which also allows us to prove the following converse result, namely Theorem 2: Suppose the quadratic fieldFis contained in some cyclic quartic extension ofQand suppose thatFhas even (wide) class number. There then is a cyclic quartic extensionKofQcontainingFsuch thatKhas no relative integral basis overF.