The Galois Group of x^2p+bx^p+c^p over ℚ

2024/01/25 by Jim, Akash, Hagedorn, Thomas
#11R09 #12E05 #12F10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2401.13925

Abstract

We prove an irreducibility criterion for polynomials of the form h(x)=x^2m + bx^m + c₁ ∈ F[x] relating to the Dickson polynomials of the first kind D_p. In the case when F = ℚ, m is a prime p>3, and c₁=c^p, for c∈ℚ, we explicitly determine the Galois group of d_h= D_p(x, c) + b, which is Aff(\mathbbF_p) or C_p \rtimes C_{(p - 1)/2} \vartriangleleft Aff(\mathbbF_p), and the Galois group of h, which is C₂ × Aff(\mathbbF_p), Aff(\mathbbF_p), or C₂ × (C_p \rtimes C_{(p - 1)/2}) \vartriangleleft C₂ × Aff(\mathbbF_p).

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