The Galois group of \(X^p+aX^{p - 1}+a\)
Let \(p\) be a prime number and \(a\in \mathbb{Z}\). The authors prove that the Galois group over \(\mathbb{Q}\) of the trinomial \(P=X^p+aX^{p-1}+a\) is the symmetric group \(S_p\) as soon as it is transitive, namely when \(a\not =\pm 1\) or \(p\not \equiv 2 \pmod 3\). The authors assume that \(P\) is irreducible, and consider the Galois group \(G\) of \(P\) over \(\mathbb{Q}\) as a transitive group of permutations of the roots of \(P\). They use a known formula to write the discriminant \(D\) of \(P\) in a certain form \(D=a_0D_0\). If \(p=3\), it is immediate that \(G=S_3\). If \(p>3\), it is known that if a prime number \(l\) divides \(D_0\) to an odd power, then \(l\) divides the discriminant of the extension \(\mathbb{Q}(\alpha)/\mathbb{Q}\) exactly once, where \(\alpha\) is a root of \(P\); in this case, it follows that \(G=S_p\), since \(G\) is transitive and contains a transposition via the inertia group of a prime ideal above \(l\) in the ring of integers of the Galois closure of \(\mathbb{Q}(\alpha)\). The strategy of the proof of the main result of the present paper consists in showing that \(D_0\) is not a square by solving some Diophantine equations.
scientific article; zbMATH DE number 5542335
scientific article