JOURNAL ARTICLE

Monogenic trinomials of the form x4+ax3+d and their Galois groups.

  • Published In: Journal of Algebra & Its Applications, 2026, v. 25, n. 10. P. 1 1 of 3

  • Database: Academic Search Ultimate 2 of 3

  • Authored By: Harrington, Joshua; Jones, Lenny 3 of 3

Abstract

Let f (x) = x 4 + a x 3 + d ∈ ℤ [ x ] , where a d ≠ 0. Let C n denote the cyclic group of order n , D 4 the dihedral group of order 8, and A 4 the alternating group of order 12. Assuming that f (x) is monogenic, we give necessary and sufficient conditions involving only a and d to determine the Galois group G of f (x) over ℚ. In particular, we show that G = D 4 if and only if (a , d) = (± 2 , 2) , and that G ∉ { C 4 , C 2 × C 2 }. Furthermore, we prove that f (x) is monogenic with G = A 4 if and only if a = 4 k and d = 2 7 k 4 + 1 , where k ≠ 0 is an integer such that 2 7 k 4 + 1 is squarefree. This paper extends previous work of the authors on the monogenicity of quartic polynomials and their Galois groups. [ABSTRACT FROM AUTHOR]

Additional Information

  • Source:Journal of Algebra & Its Applications. 2026/09, Vol. 25, Issue 10, p1
  • Document Type:Article
  • Subject Area:History
  • Publication Date:2026
  • ISSN:0219-4988
  • DOI:10.1142/S0219498826500969
  • Accession Number:193143757
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