Specializations of one-parameter families of polynomials

Farshid Hajir; Siman Wong

Displaying similar documents to “Specializations of one-parameter families of polynomials”

Algebraic properties of a family of Jacobi polynomials

John Cullinan, Farshid Hajir, Elizabeth Sell (2009)

Journal de Théorie des Nombres de Bordeaux

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The one-parameter family of polynomials $J_{n} (x, y) = \sum_{j = 0}^{n} (\binom{y + j}{j}) x^{j}$ is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each $n \geq 6$ , the polynomial $J_{n} (x, y_{0})$ is irreducible over $ℚ$ for all but finitely many $y_{0} \in ℚ$ . If $n$ is odd, then with the exception of a finite set of $y_{0}$ , the Galois group of $J_{n} (x, y_{0})$ is $S_{n}$ ; if $n$ is even, then the exceptional set is thin.

A classification of the extensions of degree $p^{2}$ over $ℚ_{p}$ whose normal closure is a $p$ -extension

Luca Caputo (2007)

Journal de Théorie des Nombres de Bordeaux

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Let $k$ be a finite extension of $ℚ_{p}$ and $ℰ_{k}$ be the set of the extensions of degree $p^{2}$ over $k$ whose normal closure is a $p$ -extension. For a fixed discriminant, we show how many extensions there are in $ℰ_{ℚ_{p}}$ with such discriminant, and we give the discriminant and the Galois group (together with its filtration of the ramification groups) of their normal closure. We show how this method can be generalized to get a classification of the extensions in $ℰ_{k}$ .

Conjugacy classes of series in positive characteristic and Witt vectors.

Sandrine Jean (2009)

Journal de Théorie des Nombres de Bordeaux

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Let $k$ be the algebraic closure of $𝔽_{p}$ and $K$ be the local field of formal power series with coefficients in $k$ . The aim of this paper is the description of the set $𝒴_{n}$ of conjugacy classes of series of order $p^{n}$ for the composition law. This work is concerned with the formal power series with coefficients in a field of characteristic $p$ which are invertible and of finite order $p^{n}$ for the composition law. In order to investigate Oort’s conjecture, I give a description of conjugacy classes of series...

PAC fields over number fields

Moshe Jarden (2006)

Journal de Théorie des Nombres de Bordeaux

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We prove that if $K$ is a number field and $N$ is a Galois extension of $ℚ$ which is not algebraically closed, then $N$ is not PAC over $K$ .

Displaying similar documents to “Specializations of one-parameter families of polynomials”

Algebraic properties of a family of Jacobi polynomials

A classification of the extensions of degree p 2 over ℚ p whose normal closure is a p -extension

Conjugacy classes of series in positive characteristic and Witt vectors.

PAC fields over number fields

A classification of the extensions of degree $p^{2}$ over $ℚ_{p}$ whose normal closure is a $p$ -extension