A construction of polynomials with squarefree discriminants

By Kiran S. Kedlaya. 2011

60 citations recorded by the source corpus.

For any integer n >= 2 and any nonnegative integers r,s with 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_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 c N^(1/(n-1)). A corollary is that for each n \geq 3, infinitely many quadratic number fields admit everywhere unramified degree n extensions whose normal closures have Galois group A_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.

Subfields: Algebra and Number Theory, Artificial Intelligence, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1f858a254852bb73c946ae87f6247e6e3b4de390

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Citations in the corpus

Squarefree values of polynomial discriminants I (2016, background): “rdering by this height, and show that the density is positive. The existence of inﬁnitely many monic integer polynomials of each degree having squarefree discriminant was ﬁrst demonstrated by Kedlaya [11]. However, it has not previously been known whether the density exists or even that the lower de”
Heights on stacks and a generalized Batyrev–Manin–Malle conjecture (2021, unknown): “One certainly expects this to be the case, but the fact, for example, that there are arbitrarily large squarefree integers which are discriminants of degree-n extensions of Q is a recent result of Kedlaya [40].”