A construction of polynomials with squarefree discriminants
Name
Kedlaya_A construction of polynomials.pdf
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256 KB
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Author(s)
Kedlaya, Kiran S.
Date Issued
September 2012
Journal
Proceedings of the American Mathematical Society
Publisher
American Mathematical Society (AMS)
Citation
Kedlaya, Kiran S. “A construction of polynomials with squarefree discriminants.” Proceedings of the American Mathematical Society 140, no. 9 (September 1, 2012): 3025-3033.
Version
Final published version
Abstract
For any integer n ≥ 2 and any nonnegative integers r, swith r+2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients having squarefree discriminant and exactly r real roots. These give rise to number fields of degree n, signature (r, s), Galois group S[subscript n], and squarefree discriminant; we may also force the discriminant to be coprime to any given integer. The number of fields produced with discriminant in the range [-N, N] is at least cN[superscript 1/(n-1)]. A corollary is that for each n ≥ 3, infinitely many quadratic number fields admit everywhere unramified degree n extensions whose normal closures have Galois group A[subscript n]. This generalizes results of Yamamura, who treats the case n = 5, and Uchida and Yamamoto, who allow general n but do not control the real place.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1090/s0002-9939-2012-11231-6
