New York Journal of Mathematics Volume 29 (2023), 171-192
 Volume 29 Volume Index

Compositum of two number fields of prime degree

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 Published: February 12, 2023. Keywords: Algebraic numbers, compositum of two number fields, transitive permutation groups of prime degree, Galois groups. Subject [2020]: 11R04, 11R32, 12F05, 20B35.

Abstract
In this paper by exploiting the properties of transitive permutation groups of prime degree we provide an answer to the following problem: given two number fields K and L both of prime degree p over Q, what values the degree of their compositum KL can take? We show that if K and L are linearly disjoint over Q, then necessarily KL has degree p2 or, for example, if K and L are number fields of prime degree p such that p=(qn-1)/(q-1) with q prime or a power of a prime, n>2, and some intermediate group between the projective special linear group PSL(n,q) and the projective semilinear group PΓL(n,q) is realizable over Q, then the degree of KL is pqn-1. In addition, for any divisor s of p-1, there exist number fields K and L of prime degree p such that their compositum KL has degree ps. As a numerical application, we determine the complete list of values the degree of compositum KL can take if K and L are two number fields of degree 13. We also give an answer to the related problem, namely, given two algebraic numbers α and β both of prime degree p, what values the degree of α+β and αβ can take?

Acknowledgements

The author is grateful to A. Dubickas for pointing out many helpful suggestions, and to
P. Drungilas for the proofs of Lemma 2.4 and Lemma 2.8. The author also thanks the referee for valuable comments.

Author information

Paulius Virbalas:
Institute of Mathematics
Faculty of Mathematics and Informatics
Vilnius University
Naugarduko 24, Vilnius LT-03225, Lithuania

paulius.virbalas@mif.stud.vu.lt