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Laguerre polynomials with Galois group \(A_m\) for each \(m\). (English) Zbl 1287.11123

Summary: In 1892, D. Hilbert [J. Reine Angew. Math. 110, 104–129 (1892; JFM 24.0087.03)] began what is now called Inverse Galois Theory by showing that for each positive integer \(m\), there exists a polynomial of degree \(m\) with rational coefficients and associated Galois group \(S_m\), the symmetric group on \(m\) letters, and there exists a polynomial of degree \(m\) with rational coefficients and associated Galois group \(A_m\), the alternating group on \(m\) letters. In the late 1920s and early 1930s, I. Schur [S.-Ber. Akad. Berlin 1930, 443–449 (1930; JFM 56.0110.02), J. Reine Angew. Math. 165, 52–58 (1931; Zbl 0002.11501; JFM 57.0125.05)] found concrete examples of such polynomials among the classical Laguerre polynomials except in the case of polynomials with Galois group \(A_m\) where \(m \equiv 2\pmod 4\). Following up on work of R. Gow from 1989 [J. Number Theory 31, 201–207 (1989; Zbl 0693.12009)], this paper complements the work of Schur by showing that for every positive integer \(m \equiv 2\pmod 4\), there is in fact a Laguerre polynomial of degree \(m\) with associated Galois group \(A_m\).

MSC:

11R09 Polynomials (irreducibility, etc.)
11R32 Galois theory
11C08 Polynomials in number theory
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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