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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.

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    $\begingroup$ Any good set of undergraduate lecture notes on Galois Theory (or a good textbook on the subject) paired with the knowledge you have now would suffice. Whether a quintic is solvable by radicals or not relies on the associated Galois group being soluble or not (for example $S_5$ isn't soluble and arises as the Galois group of some quintics over $\mathbb Q$). $\endgroup$
    – ah11950
    Commented Apr 11, 2014 at 15:28
  • $\begingroup$ @AndrewThompson Unsolvability of quintics might mean two different things. Do you mean Abel-Ruffini theorem or the more general resolution of Galois about existence of polynomials over $\Bbb Q$ that can be solved through radicals? The latter requires Galois theory and complicated field theory, but you need only basic group theory for a proof of the former, precisely Abel and Ruffini's proof presented loads before Galois' proof. $\endgroup$ Commented Apr 21, 2014 at 19:54
  • $\begingroup$ Abel-Ruffini. (Gathering what I can of mathematical national pride.) $\endgroup$ Commented Apr 21, 2014 at 21:07
  • $\begingroup$ @AndrewThompson Then you need some easy field and field extension theory and loads of group theory. $\endgroup$ Commented Apr 22, 2014 at 10:38

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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.

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