Quartic Extensions Research Articles

The Construction of Unramified Cyclic Quartic Extensions of Q(√m)

The Construction of Unramified Cyclic Quartic Extensions of Q(√m)

The construction of unramified cyclic quartic extensions of 𝑄(√𝑚)

We give an elementary general method for constructing fields K satisfying [ K : Q ] = 8 [K:Q] = 8 , the Galois group of K over Q is dihedral, and K is unramified over one of its quadratic subfields. Given an integer m, we describe all such fields K which contain Q ( m ) Q(\sqrt m ) . The description is specific and is given in terms of the arithmetic of the quadratic subfields of K.

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

K is a cyclic quartic extension of Q iff K = Q((rd + p d 1 2 ) 1 2 ) , where d > 1, p and r are rational integers, d squarefree, for which p 2 + q 2 = r 2 d for some integer q. Following a paper of A. A. Albert we show that the absolute discriminant, d( K Q ) , of the general cyclic quartic extension is given by d( K Q ) = (W 2d 2) for an explicitly computable rational integer W. We next find that the relative discriminant, d( K F ) , is given by d( K F ) = (W d 1 2 ) , where F = Q (d 1 2 ) 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 F = Q(d 1 2 ) is odd then every cyclic quartic extension K of Q containing F has a relative integral basis over F. 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 field F is contained in some cyclic quartic extension of Q and suppose that F has even (wide) class number. There then is a cyclic quartic extension K of Q containing F such that K has no relative integral basis over F.