On the degree of compositum of two number fields

Authors

  • Paulius Drungilas,

    1. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania. Phone: +37052193077, Fax: +37052151585, Phone: +37052193081, Fax: +37052151585
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  • Artūras Dubickas,

    Corresponding author
    1. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania. Phone: +37052193077, Fax: +37052151585, Phone: +37052193081, Fax: +37052151585
    • Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania. Phone: +37052193081, Fax: +37052151585
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  • Florian Luca

    1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México. Phone: +4433222777, Fax: +4433222732
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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 abc and b ≤ 6, for which there exist algebraic numbers α, β and γ of degrees a, b and c, respectively, such that α + β + γ = 0. Now, this characterization is extended up to b ≤ 7. We also solve a similar problem for equation image with ab ≤ 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 − ℓ)), where ℓ ≥ 2 is an integer and p > ℓ2 − ℓ + 1 is a prime number, is shown to be not compositum-feasible, not sum-feasible and not product-feasible.

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