Examples in Galois Theory 2 - Cubic Extensions

Let \(K\) be a field of characteristic \(\ne 2\) and \(3\). In this post we discuss the Galois group of a cubic polynomial \(f\) over \(K\). The process is not very hard but there are some quite non-trivial observations.Let \(k\) be an arbitrary field and suppose \(f(X) \in k[X]\) is separable and, i.e., \(f\) has no multiple roots in an algebraic closure, and of degree \(\ge 1\). Let \[f(X)=(X-x_1)\cdots(X-x_n)\] be its factorisation in a splitting field \(F\). Put \(G=G(L/k)\). We say that \(G...

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