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On a Converse to the Tscgebotarev density theorem

Published online by Cambridge University Press:  09 April 2009

C. E. Van Der Ploeg
C. E. Van Der Ploeg
Affiliation:
Mathematics DivisonUniversity of SussexFalmer, Brighton United Kingdom
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Abstract

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Using an elementary counting procedure on biquadratic polynomials over Zp 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 probility of separable quartic polynomials (mod p) whose roots generate numberfields with normal closure having Galois group isomorphic to D4, as p → ∞. This verifies a conjecture about a converse to the Tschebotarev density theorem. Further evidence in support of this conjecture is provided in quadratic and coubic numberfields.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 3 , June 1988 , pp. 287 - 293
Copyright
Copyright © Australian Mathematical Society 1988

References

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