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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 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 \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 − 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.

A Newman polynomial has all the coefficients in {0, 1} and constant term 1, whereas a Littlewood polynomial has all coefficients in {−1, 1}. We call P (X) ∈ Z[X] a Borwein polynomial if all its coefficients belong to {−1, 0, 1} and P (0) = 0. By exploiting an algorithm which decides whether a given monic integer polynomial with no roots on the unit circle |z| = 1 has a non-zero multiple in Z[X] with coefficients in a finite set D ⊂ Z, for every Borwein polynomial of degree at most 9 we determine whether it divides any Littlewood or Newman polynomial. In particular, we show that every Borwein polynomial of degree at most 8 which divides some Newman polynomial divides some Littlewood polynomial as well. In addition to this, for every Newman polynomial of degree at most 11, we check whether it has a Littlewood multiple, extending the previous results of Borwein, Hare, Mossinghoff, Dubickas and Jankauskas.

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Let k be a field, and let α and α be two algebraic numbers conjugate over k. We prove a result which implies that if L ⊂ k(α, α ) is an abelian or HamiltonianThis is related to a certain question concerning the degree of an algebraic number and the degree of a quotient of its two conjugates provided that the quotient is a root of unity, which was raised (and answered) earlier by Cantor. Moreover, we introduce a new notion of the non-torsion power of an algebraic number and prove that a monic polynomial in X-irreducible over a real field and having m roots of equal modulus, at least one of which is real-is a polynomial in X m .Keywords: field; subfield; algebraic numbers; abelian and Hamiltonian extensions; roots of unity 2000 Mathematics subject classification: Primary 11R04; 11R20; 11R32; 12F10 ResultsLet k be a field, and let k a be an algebraic closure of k. In this paper we give a short and self-contained proof of the following theorem.The condition of the theorem on L ∩ k(α) always holds if L/k is an abelian extension. However, this is not the only case. We say that L is a Hamiltonian extension of k if it is a Galois extension and Gal(L/k) is a Hamiltonian group, namely, a non-abelian group every subgroup of which is normal. For example, the quaternion group Q 8 is Hamiltonian. It occurs as a Galois group over the field of rational numbers Q, since it is a 2-group. See p. 190 of [8] for the description of all Hamiltonian groups.