Explicit class field theory for rational function fields

Abstract

Developing an idea of Carlitz, I show how one can describe explicitly the maximal abelian extension of the rational function field over F, (the finite field of q elements) and the action of the idèle class group via the reciprocity law homomorphism. The theory is closely analogous to the classical theory of cyclotomic extensions of the rational numbers.

Values of certain L-series in positive characteristic

“…The complete solution of the explicit class field theory problem for global fields of positive characteristic given by Hayes [24] gave impetus to the development of the theory. Shortly later, Goss [17] extended the study of of these series to the case of A = Γ(X \{∞}, O X ), where X is a smooth projective curve defined over F q and ∞ is a closed point of X. Goss also provided the necessary analytic structure to study these series.…”

Global Function Fields With Many Rational Places Over the Quinary Field

“…Moreover, every place of Hp lying over Q is totally ramified in EM/Hp. In the special case where K = F,(x), the theory of narrow ray class extensions reduces to that of cyclotomic function fields as developed by Hayes [3]. We note that cyclotomic function fields and narrow ray class extensions have already been used by Niederreiter and Xing [6], [8], [9], Quebbemann [12], and Xing and Niederreiter [18], [19] for the construction of global function fields with many rational places.…”

On coefficient valuations of Eisenstein polynomials

“…We want to study a function field analogue over Q of the number field extension Q(ζ p n )|Q. Since 1 is the only p n th root of unity in an algebraic closureQ, we have to proceed differently, following Carlitz [1] and Hayes [5]. First of all, the power operation of p n onQ becomes replaced by a module operation of f n onQ, where f ∈ Z is an irreducible polynomial.…”