Type: Article

On the degree of compositum of two number fields

Drungilas, Paulius; Dubickas, Arturas; Luca, Florian;

Abstract:

We prove that the sum of two algebraic numbers both of degree 6 cannot be of degree 8. The triplet (6, 6, 8) was the only undecided case in the previous characterization of all positive integer triplets (a, b, c), with a b c and b 6, for which there exist algebraic numbers , and of degrees a, b and c, respectively, such that + + = 0. Now, this characterization is extended up to b 7. We also solve a similar problem for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(a,b,c) \in \mathbb N<^>3$\end{document} with a b 7 by finding for which positive integers a, b, c there exist number fields of degrees a and b such that their compositum has degree c. We show that the problem of describing all compositum-feasible triplets can be reduced to some finite computation in transitive permutation groups of degree c which occur as Galois groups of irreducible polynomials of degree c. In particular, by exploiting the properties of the projective special linear group PSL(2, 7) of order 168, we prove that the triplet (7, 7, 28) is compositum-feasible. In the opposite direction, the triplet (p, p, p(p )), where 2 is an integer and p > 2 + 1 is a prime number, is shown to be not compositum-feasible, not sum-feasible and not product-feasible. (C) 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim