## Publications

Abstract:

Let F, K and L be algebraic number fields such that {Mathematical expression}, [K:F]=2 and [L:K]=n. It is a simple consequence of the class field theory that, if L is an abelian, unramified extension of K and (n,h)=1, where h is the class number of F, then L is normal over F. The purpose of this note is to point out the necessity of the condition (n,h)=1 by constructing for any field F with even class number a tower of fields {Mathematical expression} with [K:F]=2, [L:K]=2 where L is unramified over K, but L is not normal over F. © 1980 Springer-Verlag.

Abstract:

Emil Artin studied quadratic extensions of k(x) where k is a prime field of odd characteristic. He showed that there are only finitely many such extensions in which the ideal class group has exponent two and the infinite prime does not decompose. The main result of this paper is: If K is a quadratic imaginary extension of k(x) of genus G, where k is a finite field of order q, in which the infinite prime of k(x) ramifies, and if the ideal class group has exponent 2, then q = 9, 7, 5, 4, 3, or 2 and G ≤ 1, 1, 2, 2, 4, and 8, respectively. The method of Artin's proof gives G ≤ 13, 9, and 9724 for q = 7, 5, and 3, respectively. If the infinite prime is inert in K, both the methods of this paper and Artin's methods give bounds on the genus that are roughly double those in the ramified case. © 1977.

Abstract:

Let Kq denote the finite field with q elements and characteristic p. Let f(x) be a monic polynomial of degree d with coeficients in Kq. Let C(f) denote the number of distinct values of f(x) as x ranges over Kq. We easily see that C(f)≥ q-1 d+1 where [{norm of matrix}] is the greatest integer function. A polynomial for which equality in (*) occurs is called a minimum value set polynomial. There is a complete characterization of minimum value set polynomials over arbitrary finite fields with d 4 15 then f(x) is one of the following polynomial forms: 1. (a) f(x) = (x + a)^{d} + b, where d | (q - 1), 2. (b) f(x) = ((x + a)^{ d 2} + b)^{2} + c, where d | (q^{2} - 1), 3. (c) f(x) = ((x + a)^{2} + b)^{ d 2} + c, where d | (q^{2} - 1), or 4. (d) f(x) = Dd,a(x + b) + c, where Dd,a(x) is the Dickson polynomial of degree d, d | (q^{2} - 1) and a is a 2^{k}th power in Kq^{2} where d = 2^{k}r, r is odd. The result is obtained by noticing the connection between the size of the value set of a polynomial f(x) and the factorization of the associated substitution polynomial f^{*}(x, y) = f(x) - f(y) in the ring Kq[x, y]. Essentually, we show that C(f) *(x, y) has at least d 2 factors in Kq[x, y], and we determine all the polynomials with such characteristic. © 1988.

Abstract:

This paper describes a method of constructing an unlimited number of infinite families of continued fraction expansions of the square root of D, an integer. The periods of these continued fractions all have identifiable sub patterns repeated a number of times according to certain parameters. For example, it is possible to construct an explicit family for the square root of D(k, l) where the period of the continued fraction has length 2kl - 2. The method is recursive and additional parameters controlling the length can be added.