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Generic polynomial
About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.
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TL;DR: In this article, it was shown that the arboreal Galois representation attached to a large class of quadratic polynomials defined over a field of rational functions in characteristic zero has finite index in the full automorphism group of the associated preimage tree.
Abstract: We prove that the arboreal Galois representation attached to a large class of quadratic polynomials defined over a field of rational functions in characteristic zero has finite index in the full automorphism group of the associated preimage tree. Moreover, we show that the index is bounded, in most cases, independently of the polynomial.
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TL;DR: The number of f(x)@?s having a given subgroup as Galois group is determined using the inequality of the following type: For α ≥ 1, β ≥ 1 using LaSalle's inequality.
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TL;DR: In this article, Borevich showed that Λ-module E has a system of n+1 generators, of which n−1 are free and two are connected by certain relations.
Abstract: Suppose k is a local field that is an extension of the field of p -adic numbers of degree n and does not contain a primitive p -th root of 1, and suppose K/k is a cyclic p-extension with Galois group G. The group E of principal units of K is a multiplicatively written module over the group ring Λ=ℤp[G], where ℤp is the ring of p-adic integers. It was shown by Borevich (Ref. Zh. Mat., 1965, 3A256) that the Λ-module E has a system of n+1 generators, of which n−1 are free and two are connected by certain relations. In the present paper these Λ-generators are constructed explicitly and their arithmetical characteristics indicated.
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TL;DR: In this article, a general framework for rigid deformations of (projective) representations of the absolute Galois group of a function field (in one variable) over a separably closed base is provided.
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