Elementary characterization of the Galois groups of x 12 + ax 6 + b

Let f ( x ) = x 12 + a x 6 + b ∈ Q [ x ] f(x)=x¹²+ax⁶+b∈ Q[x] be an irreducible polynomial, g 4 ( x ) = x 4 + ax 2 + b , g 6 ( x ) = x 6 + ax 3 + b , and let G 4 and G 6 be the Galois groups of g 4 ( x ) and g 6 ( x ), respectively. This paper provides a streamlined elementary characterization of all 16 possible Galois groups of f ( x ). In particular, we give explicit criteria showing that the Galois group of f ( x ) can be uniquely determined by the pair ( G 4 , G 6 ) along with testing whether at most two expressions involving a and b are rational squares. This contributes to a broader program to systematically study the Galois groups of x 2 m + ax m + b for a positive integer m .