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Quadratic reciprocity in a finite group
 Amer. Math. Monthly
"... In memory of Abe Hillman The law of quadratic reciprocity is a gem from number theory. In this article we show that it has a natural interpretation that can be generalized to an arbitrary finite group. Our treatment relies almost exclusively on concepts ..."
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In memory of Abe Hillman The law of quadratic reciprocity is a gem from number theory. In this article we show that it has a natural interpretation that can be generalized to an arbitrary finite group. Our treatment relies almost exclusively on concepts
The Classification of the Finite Simple Groups: An Overview
 MONOGRAFÍAS DE LA REAL ACADEMIA DE CIENCIAS DE ZARAGOZA. 26: 89–104, (2004)
, 2004
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, 2008
"... In the process of computing the Galois group of a prime degree polynomial f(x) over Q we suggest a preliminary checking for the existence of nonreal roots. If f(x) has nonreal roots, then combining a 1871 result of Jordan and the classification of transitive groups of prime degree which follows fr ..."
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In the process of computing the Galois group of a prime degree polynomial f(x) over Q we suggest a preliminary checking for the existence of nonreal roots. If f(x) has nonreal roots, then combining a 1871 result of Jordan and the classification of transitive groups of prime degree which follows from CFSG we get that the Galois group of f(x) contains Ap or is one of a short list. Let f(x) ∈ Q[x] be an irreducible polynomial of prime degree p ≥ 5 and r = 2s be the number of nonreal roots of f(x). We show that if s satisfies s(s log s + 2log s + 3) ≤ p then Gal(f) = Ap, Sp. 1