Vol. 112, No. 2, 1984

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ISSN: 0030-8730
The nonregular analogue of Tchebotarev’s theorem

Michael Fried

Vol. 112 (1984), No. 2, 303–311

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

( p  )

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.

Mathematical Subject Classification 2000
Primary: 11R37
Secondary: 11R45, 11R58
Received: 20 May 1982
Published: 1 June 1984
Michael Fried
Montana State University