0

My background is an undergrad year long course in algebra which got me through to some basic finite field and field extension stuff (most of fraliegh or Ian Stewart's book are familiar to me if that helps) but no Galois theory.

My goal is to get a good grasp on where the cubic and quartic equations come from, and as much understanding of the proof for an equation for the quintic being imposible as I can get given my background. If Galois theory is crucial, good references for concise introductions to this given my goal would be much appreciated.

operatorerror
  • 29,881
  • 1
    Just a heads up, for the last part, Galois theory is definitely crucial. – Chill2Macht Apr 24 '16 at 21:05
  • Before Galois developped his theory, the question whether quintics are solvable in general was open. – Peter Apr 24 '16 at 21:06
  • hm ok, is it something you can understand after a full course in galois theory or does it just involve some of the intro machinery? – operatorerror Apr 24 '16 at 21:06
  • I would say mostly the intro machinery, at least the proof I saw came down mostly to the theory of solvable groups and classifying finite groups. – Chill2Macht Apr 24 '16 at 21:07
  • 2
    It is not easy to follow the proof, but what you need is : A polynomial over a field with characteristic $0$ is solvable by radicals, if and only if the galois group of the polynomial over the field is solvable. The smallest non-solvable finite group is $A_5$ , so you can deduce that polynomials with degree $d\le 4$ are always solvable by radicals. – Peter Apr 24 '16 at 21:08
  • 1
    If you choose a random polynomial with integer coefficients and degree $d\ge 5$, the probability is high that the galois group of this polynomial over $\mathbb Q$ is $S_d$ and the polynomial is not solveable by radicals. – Peter Apr 24 '16 at 21:12
  • 2
    In fact, in a sense that can be made precise, the probability is $1$; see http://mathoverflow.net/questions/58397/the-galois-group-of-a-random-polynomial – Travis Willse Apr 24 '16 at 22:09

1 Answers1

1

Try the book Galois Theory for Beginners: A Historical Perspective by Bewersdorff.

Read an MAA review here.

lhf
  • 221,500