Timeline for Link between Irreducible Factors and Prime Factors (or Cycles of a Permutation)

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Jan 19, 2017 at 22:31	comment	added	Ofir Gorodetsky		(cont.) in fact, some sort of Artin symbol. Chebotarev tells us that the this Artin symbol is equidistributed in the Galois group of $P(T,a_0,\cdots,a_{n-1})$ as $q \to \infty$. See, for instance Proposition 3.1 in the paper by Bank et. al: arxiv.org/pdf/1302.0625v3.pdf . Technically they apply Lang-Weil, but it is Chebotarev in disguise.
Jan 19, 2017 at 22:29	comment	added	Ofir Gorodetsky		I like your post! Regarding the density theorem - I want to elaborate, although I am not an expert. Let $a_0, \cdots, a_{n-1}$ be $n$ variables and consider the polynomial $P(T,a_0,\cdots,a_{n-1}) = T^n + \sum a_i T^i$ - the generic monic polynomial of degree $n$. Its Galois group over $\overline{\mathbb{F}_q}(a_0,\cdots,a_{n-1})$ is $S_n$. For each specialization of $a_0,\cdots,a_{n-1}$ we get a polynomial, whose factorization may be read from the action of the Frobenius. The function that associates to each specialization its Frobenius-induced permutation (up to conjugaction) is...
Jan 19, 2017 at 21:57	history	answered	Qiaochu Yuan	CC BY-SA 3.0