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- Title
Galois groups of certain even octic polynomials.
- Authors
Chen, Malcolm Hoong Wai; Chin, Angelina Yan Mui; Tan, Ta Sheng
- Abstract
Let f (x) = x 8 + a x 4 + b ∈ ℚ [ x ] be an irreducible polynomial where b is a square. We give a method that completely describes the factorization patterns of a linear resolvent of f (x) using simple arithmetic conditions on a and b. As a result, we determine the exact six possible Galois groups of f (x) and completely classify all of them. As an application, we characterize the Galois groups of irreducible polynomials x 8 + a x 4 + 1 ∈ ℚ [ x ]. We also use similar methods to obtain analogous results for the Galois groups of irreducible polynomials x 8 + a x 6 + b x 4 + a x 2 + 1 ∈ ℚ [ x ].
- Subjects
IRREDUCIBLE polynomials; POLYNOMIALS; ARITHMETIC; RESOLVENTS (Mathematics)
- Publication
Journal of Algebra & Its Applications, 2023, Vol 22, Issue 12, p1
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S0219498823502638