A couple of remarks in Galois Theory

Very short post today. Here are a couple of potentially useful Galois Theory-related things that I've discovered doing the 2005 past papers.

• It's quite hard to distinguish between $D_8$ and $Q_8$ as groups, e.g. when trying to find the Galois group of a polynomial. Note that $Q_8$ only has one element of order 2, and it commutes with everything. If your group is going to be $D_8$, there's a good chance that complex conjugation won't commute with stuff, if it's your 2-cycle.
• If you're trying to analyse algebraic relations in the Galois group, you can use transitivity to produce explicit examples of automorphisms (in general it's quite hard to state an automorphism other than obvious conjugation explicitly). Just knowing that $\sigma$ sends $x_1$ to $x_2$ might be enough to help you out (e.g. when you need something that doesn't commute with complex conjugation). 
• Critically, remember the (basic) fact that if you've proven your polynomial is irreducible, then it is the minimal polynomial for each of its roots. (Don't make the mistake of assuming this in general!) In particular, we know (for free) the dimension of the extension obtained by adjoining any one of the roots.