# 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...