Lithuanian Mathematical Journal

, Volume 59, Issue 1, pp 39–47

# A degree problem for the compositum of two number fields*

• Paulius Drungilas
• Lukas Maciulevičius
Article

## Abstract

The triplet (a, b, c) of positive integers is said to be compositum-feasible if there exist number fields K and L of degrees a and b, respectively, such that the degree of their compositum KL equals c. We determine all compositum-feasible triplets (a, b, c) satisfying ab and b ∈ {8, 9}. This extends the classification of compositum-feasible triplets started by Drungilas, Dubickas, and Smyth [5]. Moreover, we obtain several results related to triplets of the form (a, a, c). In particular, we prove that the triplet (n, n, n(n − 2)) is not compositum-feasible, provided that n ≥ 5 is an odd integer.

## Keywords

algebraic number sum-feasible product-feasible compositum-feasible

11R04 11R32

## References

1. 1.
G. Baron, M. Drmota, and M. Skałba, Polynomial relations between polynomial roots, J. Algebra, 177(3):827–846, 1995.
2. 2.
J.D. Dixon and B. Mortimer, Permutation Groups, Grad. Texts Math, Vol. 163, Springer, New York, 1996.Google Scholar
3. 3.
P. Drungilas and A. Dubickas, On degrees of three algebraic numbers with zero sum or unit product, Colloq. Math., 143(2):159–167, 2016.
4. 4.
P. Drungilas, A. Dubickas, and F. Luca, On the degree of the compositum of two number fields, Math. Nachr., 286(2–3):171–180, 2013.
5. 5.
P. Drungilas, A. Dubickas, and C.J. Smyth, A degree problem of two algebraic numbers and their sum, Publ. Math., 56(2):413–448, 2012.
6. 6.
I.M. Isaacs, Degrees of sums in a separable field extension, Proc. Am. Math. Soc., 25:638–641, 1970.
7. 7.
C.U. Jensen, A. Ledet, and N. Yui, Generic Polynomials. Constructive Aspects of the Inverse Galois Problem, Cambridge Univ. Press, Cambridge, 2002.
8. 8.
S. Lang, Algebra, 3rd revised ed., Grad. Texts Math., Vol. 211, Springer, New York, 2002.Google Scholar
9. 9.
A.R. Perlis, Roots appear in quanta, Am. Math. Monthly, 111(1):61–63, 2004.
10. 10.
C.J. Smyth, Additive and multiplicative relations connecting conjugate algebraic numbers, J. Number Theory, 23:243–254, 1986.
11. 11.
I. Stewart, Galois Theory, 4th ed., Chapman & Hall/CRC, Boca Raton, FL, 2015.