Galois Theory

Galois theory provides an intrinsic explanation for why the roots of a general polynomial of degree 5 or higher cannot be expressed as nested radical expressions based on the coefficients of the polynomial. Let's do a quick review of the quadratic, bi-quadratic and Cardano/Tartaglia formulas. We'll look at some of the consequences of these formulas (e.g.) discriminants and the origins of algebraic number theory. Then I suggest diving right into the main theorem of Galois Theory: the one-to-one, lattice-inverting correspondence between subfields (of a Galois extension) and subgroups (of the Galois group). I'd like to do a lot of examples. It's always fun to calculate. The aim is to make this incredibly powerful and beautiful theory accessible to interested people of any background. Or as Galois described it: déchiffrer tout ce gâchis (decipher all this mess). It is going to be a conversation about real mathematics, however. That means you should be comfortable with the pattern: example, abstraction, definition, lemma/theorem/corollary, proof.