Galois Group Computation for Rational Polynomials
نویسندگان
چکیده
منابع مشابه
Galois Group Computation for Rational Polynomials
We describe methods for the computation of Galois groups of univariate polynomials over the rationals which we have implemented up to degree 15. These methods are based on Stauduhar's algorithm. All computations are done in unramiied p-adic extensions. For imprimitive groups we give an improvement using subbelds. In the primitive case we use known subgroups of the Galois group together with a c...
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In 1892, D. Hilbert 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 Sm, the symmetric group on m letters, and there exists a polynomial of degree m with rational coefficients and associated Galois group Am, the alternating group on m letters. In the late 1920’...
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Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed α ∈ Q−Z<0, Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L (α) n (x) = ∑n j=0 ( n+α n−j ) (−x)/j! is irreducible for all large enough n. We use our criterion to show that, under these conditions, the Galois group of L (α) n (x) is...
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This motivating question is lifted almost verbatim from B. Wyman’s 1972 article, [Wym72], of which the present article is merely an updated version. It may surprise the reader to learn that a large swath of modern number theory known as the Langlands program is dedicated to variations on the theme of Question A. We ought to clarify what is meant by a “rule” in Question A. We are not looking for...
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ژورنال
عنوان ژورنال: Journal of Symbolic Computation
سال: 2000
ISSN: 0747-7171
DOI: 10.1006/jsco.2000.0377