Kenneth Hardy

Determination of all imaginary cyclic quartic fields with class number 2

It is proved that there are exactly 8 imaginary cyclic quartic fields with class number 2.

Calculation of the class numbers of imaginary cyclic quartic fields

Any imaginary cyclic quartic field can be expressed uniquely in the form K = Q(\JA(D + b/d) ), where A is squarefree, odd and negative, D = B2 + C2 is squarefree, B > 0, C > 0, and (A,D)= 1. Explicit formulae for the discriminant and conductor of K are given in terms of A, B, C, D. The calculation of tables of the class numbers h(K) of such fields K is described. Let Q denote the field of rational numbers and let K be a cyclic extension of Q of degree 4. The unique quadratic subfield of K is denoted by k. The class number of K (resp. k) is denoted by h(K) (resp. h(k)). The conductor of K is denoted by / = /( K ). In the case of real cyclic quartic fields K, Gras (3) has given a table of the values of h(K) for all such fields with / < 4000. Recently, the authors have carried out the calculation of the class numbers of imaginary cyclic quartic fields (4). In this note we give a brief description of the computation of the tables given in (4). The following explicit representation of a cyclic quartic field is proved in (4,

A refinement of h. c. williams?th root algorithm

Let p and q be primes such that p =1 (mod q). Let a be an integer such that a(P-l)/q = 1 (mod p). In 1972, H. C. Williams gave an algorithm which determines a solution of the congruence Xq _ a (mod p) in O(q3 logp) steps, once an integer b has been found such that (bq a)(P1)/q 0 0, 1 (mod p) . A step is an arithmetic operation (mod p) or an arithmetic operation on q-bit integers. We present a refinement of this algorithm which determines a solution in O(q4)+O(q2 logp) steps, once b has been determined. Thus the new algorithm is better when q is small compared with p .

On normal functions

Abstract Dilworth defined normal (upper semicontinuous) functions in [2] and used them to describe the Dedekind completion of C∗(X), the lattice of bounded continuous real-valued functions on a completely regular Hausdorff space X. In this paper we present some new properties of normal functions which enable us to view certain sets of normal functions defined on a Baire space as direct limits of sets of continuous functions. This provides various operations on normal functions and, in turn, a connection with some of the rings studied by Fine et al. in [3].

Congruences modulo 16 for the class numbers of complex quadratic fields

Abstract Let h ( d ) denote the class number of the quadratic field Q (√ d ) of discriminant d . A number of new determinations of h ( d ) modulo 16 are proved. For example, it is shown that if p and q are primes satisfying p ≡ q ≡ 5 (mod 8), ( p q ) = 1 , then h(−8pq)≡ 4( mod 16) if aA+bB p =(−1) (b+B+4) 4 12( mod 16) if aA+bB p =(−1) (b+B) 4 where a and b are unique integers such that p = a 2 + b 2 , a ≡ 1 (mod 4), b ≡ ( (p − 1) 2 )! a ( mod p) , and A and B are the unique integers such that q = A 2 + B 2 , A ≡ 1 (mod 4), B ≡ ( (q − 1) 2 )! A ( mod q) .