Let L∕K be a Galois
extension of function fields in one variable where K has exact constants F(q), the
finite field with q elements. For l a fixed integer and C a conjugacy class of
G(L∕K), this paper counts the primes p of K of degree l for which the Artin
symbol
equals C (Theorem 1.4). The answer depends on the restriction of elements
of C to the algebraic closure of F(q) in L: a proper extension of F(q) in
general.
For l = 1 [Fr; Proposition 2] followed Dirichlet’s celebrated argument using the
rationality of L-series. Tchebotarev’s original “field crossing argument” [T] is a part of
the reduction to the cyclic case ([D] and [M]) that at once removes the restriction on
l and the need for L-series (other than the Riemann hypothesis for curves over finite
fields). This more elementary argument also improves the error estimates and
therefore such practical applications as [FrS] and explicit forms of Hilbert’s
irreducibility theorem [Fr; §3]. We comment briefly on the latter (§2) to facilitate its
use in [Fr, 2; §4] for the explicit production of rank 12 elliptic curves over
Q.
