Carlitz-Kummer function fields

Abstract

Let F q be a finite field with q elements where q is a power of a prime p. Also, let M be any polynomial in F q [ x] and let k M be the Mth cyclotomic function field. If K is any finite extension field of k M , then we define Carlitz-Kummer extensions of K as an analog of the Kummer extensions of algebraic number fields. More specifically, let z ∈ K. Then the Carlitz-Kummer extension K M, z is defined as the splitting field over K of the polynomial u M − z. The Carlitz-Kummer extension K M, z is a simple, separable, Abelian extension whose degree is a power of the characteristic. Our main results are on the factorization of primes in Carlitz-Kummer extensions. Let q be any prime divisor of K and let Q be any prime divisor of K M, z that lies over q . We show that q can ramify in K M, z only if q is an infinite prime, q divides M, or q divides the denominator of z. Finally, we show that the factorization of q in K M, z in determined by certain congruence conditions on the polynomial u M − z modulo powers of the prime q .