I am currently doing a course in Abstract Algebra. I have been told that while some of the basic theory is laid down, we will not get as far as actually proving the unsolvability of quintics. Therefore, I ask you to point me in a direction (in terms of books, articles) where I can obtain the necessary knowledge. You may assume that I already have knowledge of the following:
- Basic group theory (Definition of a group, cyclic groups, symmetric groups, theorem of finitely generated abelian groups, group actions, Burnside etc.)
- Basic ring theory (Definition of rings, integral domains, fields, irreducible polynomials, algebraic and transcendental elements, Field extensions etc.)
- Very basic understanding of the impossibility of certain geometric constructions, such as trisecting the angle, doubling the cube, squaring the circle etc.
Any book/article-recommendations that do not require (much) more knowledge than what I have listed above would be highly appreciated.
Herstein's Topics in Algebra gives the proof in chapter 5, and will also teach you what you need to know about fields to follow the proof.