Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree

Virbalas, Paulius

doi:10.3390/math11061485

Open AccessArticle

Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree

by

Paulius Virbalas

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

Mathematics 2023, 11(6), 1485; https://doi.org/10.3390/math11061485

Submission received: 2 February 2023 / Revised: 15 March 2023 / Accepted: 16 March 2023 / Published: 18 March 2023

(This article belongs to the Section A: Algebra and Logic)

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Abstract

Let

α

and

β

be two algebraic numbers such that

deg (α) = m

and

deg (β) = p

, where p is a prime number not dividing m. This research is focused on the following two objectives: to discover new conditions under which

deg (α β) = m p

; to determine the complete list of values

deg (α β)

can take. With respect to the first question, we find that if the minimal polynomial of

β

over

Q

is neither

x^{p} + c

nor

x^{2} + c x + c^{2}

, then necessarily

deg (α β) = m p

and

α β

is a primitive element of

Q (α, β)

. This supplements some earlier results by Weintraub. With respect to the second question, we determine that if

p > 2

and

p - 1

divides m, then for every divisor k of

p - 1

, there exist

α

and

β

such that

deg (α β) = m p / k

.

Keywords: degree of an algebraic number; Galois theory; transitive permutation groups

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MDPI and ACS Style

Virbalas, P. Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree. Mathematics 2023, 11, 1485. https://doi.org/10.3390/math11061485

AMA Style

Virbalas P. Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree. Mathematics. 2023; 11(6):1485. https://doi.org/10.3390/math11061485

Chicago/Turabian Style

Virbalas, Paulius. 2023. "Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree" Mathematics 11, no. 6: 1485. https://doi.org/10.3390/math11061485

APA Style

Virbalas, P. (2023). Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree. Mathematics, 11(6), 1485. https://doi.org/10.3390/math11061485

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Degree of the Product of Two Algebraic Numbers One of Which Is of Prime Degree

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