Monogenity in totally real extensions of imaginary quadratic fields with an application to simplest quartic fields

Quadratic extensions of totally real quintic fields

In this work, we establish lists for each signature of tenth degree number fields containing a totally real quintic subfield and of discriminant less than 1013 in absolute value. For each field in the list we give its discriminant, the discriminant of its subfield, a relative polynomial generating the field over one of its subfields, the corresponding polynomial over Q, and the Galois group of ...

Zp-Extensions of Imaginary Quadratic Fields

Class number in totally imaginary extensions of totally real function fields

We show that, up to isomorphism, there are only finitely many totally real function fields which have any totally imaginary extension of a given ideal class number.

Zp-Extensions of Totally Real Fields

We continue our investigations into complex and p-adic variants of H. M. Stark’s conjectures [St] for an abelian extension of number fields K/k. We have formulated versions of these conjectures at s = 1 using so-called ‘twisted zeta-functions’ (attached to additive characters) to replace the more usual L-functions. The complex version of the conjecture was given in [So3]. In [So4] we formulated...

Visualizing imaginary quadratic fields

Imaginary quadratic fields Q( √ −d), for integers d > 0, are perhaps the simplest number fields afterQ. They are equal parts helpful first example and misleading special case. LikeZ, the Gaussian integersZ[i] (the cased = 1) have unique factorization and a Euclidean algorithm. As d grows, however, these properties eventually fail, first the latter and then the former. The classical Euclidean al...