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Number Theorymathscidoc:1610.24002

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let l be a rational prime and K a rational function field F_q(t) with l\nmid q. Let Discf (F/K) denote the finite discriminant of F over K. Denote the number of abelian l-extensions F/K with deg (Discf (F/K)) = (l − 1)αn by a_l(n), where α = α(q, l) is the order of q in the multiplicative group (Z/lZ)^\times. We give a explicit asymptotic formula for a_l(n). In the case of cubic extensions with q ≡ 2 (mod 3), our formula gives an exact analogue of Cohn’s classical formula.