Galois groups over rational function fields and explicit Hilbert irreducibility
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866929225033842688 |
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| author | Krumm, David Sutherland, Nicole |
| author_facet | Krumm, David Sutherland, Nicole |
| contents | Let $P\in\mathbb Q[t,x]$ be a polynomial in two variables with rational coefficients, and let $G$ be the Galois group of $P$ over the field $\mathbb Q(t)$. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $c$ the specialized polynomial $P(c,x)$ has Galois group isomorphic to $G$ and factors in the same way as $P$. In this paper we discuss methods for computing the group $G$ and obtaining an explicit description of the exceptional numbers $c$, i.e., those for which $P(c,x)$ has Galois group different from $G$ or factors differently from $P$. To illustrate the methods we determine the exceptional specializations of three sample polynomials. In addition, we apply our techniques to prove a new result in arithmetic dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1708_04932 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Galois groups over rational function fields and explicit Hilbert irreducibility Krumm, David Sutherland, Nicole Number Theory Let $P\in\mathbb Q[t,x]$ be a polynomial in two variables with rational coefficients, and let $G$ be the Galois group of $P$ over the field $\mathbb Q(t)$. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $c$ the specialized polynomial $P(c,x)$ has Galois group isomorphic to $G$ and factors in the same way as $P$. In this paper we discuss methods for computing the group $G$ and obtaining an explicit description of the exceptional numbers $c$, i.e., those for which $P(c,x)$ has Galois group different from $G$ or factors differently from $P$. To illustrate the methods we determine the exceptional specializations of three sample polynomials. In addition, we apply our techniques to prove a new result in arithmetic dynamics. |
| title | Galois groups over rational function fields and explicit Hilbert irreducibility |
| topic | Number Theory |
| url | https://arxiv.org/abs/1708.04932 |