J Austral Math Soc Ser A 44 pp287--293, 1988.

On a Converse to the Tschebotarev Density Theorem

C. E. van der Ploeg

Abstract

Using an elementary counting procedure on biquadratic polynomials over Z_p it is shown that the probability distribution of odd, unramified rational primes according to decomposition type in a fixed dihedral numberfield is identical to the probability distribution of separable quartic polynomials (mod p) whose roots generate numberfields with a normal closure having Galois group isomorphic to D₄, as p ® ¥. This verifies a conjecture about a converse to the Tschebotarev density theorem. Further evidence is support of this conjecture is provided in quadratic and cubic numberfields.

1980 AMS Subject Classification: 12A30

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Authors

C. E. van der Ploeg

Mathematics Division, University of Sussex, Falmer, Brighton, United Kingdom.