Discussion
What is the Galois group of \(f(x) = x^{3} - 3x^{2} + 15x - 3\) over Q ?
Answer: D
The Galois group is the permutation group \(S_{3}\).
The polynomial is irreducible over Q by Eisenstein's criterion at 3: 3 divides \(-3\), 15 and \(-3\), 3 does not divide the leading coefficient 1, and 9 does not divide the constant \(-3\). So the group is a transitive subgroup of \(S_{3}\), either \(A_{3} \cong \mathbf{Z}_{3}\) or \(S_{3}\).
The derivative \(3x^{2} - 6x + 15 = 3[(x - 1)^{2} + 4]\) is always positive, so f is increasing and has exactly one real root and two complex roots. Complex conjugation then swaps two roots and fixes one, a transposition, which is not in \(A_{3}\). Hence the group is \(S_{3}\). (Equivalently, the discriminant is \(-9612 \lt 0\), not a square.)
\(\mathbf{Z}_{4}\) and \(A_{4}\) cannot be Galois groups of a cubic, since the group of a cubic sits inside \(S_{3}\), of order 6.
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