Elementary characterization for Galois groups of x¹²+ax⁶+b

2024/10/01 by Malcolm Hoong Wai Chen, Chen, Malcolm Hoong Wai
Mathematics · #11R09 #12-08 #12D05 #12F10 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Characterization (materials science) #Commutative Algebra and Its Applications #FOS: Mathematics #Galois group #Mathematics #Number Theory (math.NT) #Optics #Physics #Pure mathematics

paper · pdf · doi:10.48550/arxiv.2410.00870

openalex publication_date 2024/10/01 · openalex created_date 2024/10/29 · openalex updated_date 2026/07/28

Abstract

Let f(x)=x¹²+ax⁶+b ∈ ℚ[x] be an irreducible polynomial, g₄(x)=x⁴+ax²+b, g₆(x)=x⁶+ax³+b, and let G₄ and G₆ be the Galois group of g₄(x) and g₆(x), respectively. Building upon known characterizations of G₄ and G₆ in the literature, this paper provides an elementary characterization of all sixteen possible Galois groups of f(x). In particular, we show that the Galois group of f(x) can be uniquely determined by the pair (G₄,G₆) along with testing whether at most two expressions involving a and b are rational squares.

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