Charles J. Parry

The fermat equation over quadratic fields

Abstract Kummers method of proof is applied to the Fermat equation over quadratic fields. The concept of an m-regular prime, p, is introduced and it is shown that for certain values of m, the Fermat equation with exponent p has no nontrivial solutions over the field Q(√m).

Units of algebraic numberfields

Abstract Let K be an algebraic number field with proper subfield k . If K and k have the same number of fundamental units then relations between the units of K and k are obtained.

On relative integral bases for cyclic quartic fields

Abstract When does a cyclic quartic field have an integral basis over its quadratic subfield? A simple, easy to use answer is given to this question. Moreover, a basis is given whenever it exists.

Units of modulus 1

Abstract If k is an algebraic number field which is normal over the field of rational numbers then it is shown that k has nontrivial units of modulus 1 if and only if the maximal real subfield of k is also a normal extension of the rationals. A characterization of the units is given for fields which satisfy the above conditions. A new proof of Kummers Theorem on the units of cyclotomic fields is also obtained.

Steinitz classes of order 2 in quadratic and quartic fields

Abstract For a given number field K, does there exist an extension M of odd prime degree l such that the relative discriminant of M/K is a principal ideal, but M/K has no relative integral basis? A general, but incomplete answer is given to this question when K/Q is a normal extension. If, in addition, [K:Q] is odd, the answer is complete. A detailed study is done when K/Q is a quadratic or normal quartic extension.

Primes represented by binary quadratic forms

Abstract In 1882 Weber showed that any primitive binary quadratic form with integral coefficients represents infinitely many primes in any arithmetic progression consistent with the generic characters of the form. In this paper it is shown that for any two primitive integral binary quadratic forms with unequal but fundamental discriminants, there is an infinite set of prime numbers p in any arithmetic progression consistent with the generic characters of the forms such that both forms represent p .