Fields and Galois Theory (Springer Undergraduate Mathematics Series) [Paperback]

This is a short but very good introductory book on abstract algebra, with emphasis on Galois Theory. Very little background in mathematics is required, so that the potential audience for this book range from undergraduate and graduate students, researchers, computer professionals, and the math enthusiasts.

Have you ever wondered why there are no general formulas for the roots of quintic or higher-degree polynomials with rational coefficients which involve only addition, subtraction, multiplication, division, and taking roots of the coefficents? Who hasn't? What, you say you haven't? Well, if you remember back to high-school algebra and the good old quadratic equation ax^2 + bx + c = 0, there is a solution x = -b plus or minus square root of b^2 minus 4ac, all over 2a. Turns out there is also a formula, albeit more complicated, for the cubic equation ax^3 + bx^2 + cx + d = 0, and an even more complicated one for the quartic equation ax^4 + bx^3 + cx^2 + dx + e = 0. But there's no such general formula for the roots of quintic (x^5) or higher polynomials with rational coefficients. The two mathematicians who proved this were both doomed to die at an early age: Niels Abel, a Norwegian who died of tuberculosis at age 27, and Evariste Galois, a Frenchman who died of a bullet in the gut received in a duel at age 20. (Ironically, no one remembers who or what they were fighting over. I hope it was worth it!)

If you'd like to know why this is so, this book will get you there. I've had a fair amount of exposure to higher math, so I'm not sure I can accurately determine whether someone with only high-school algebra could follow this book, but I think the answer would be a qualified "yes", assuming there was sufficient motivation and persistence. In addition to high-school algebra, the only other background someone would need is a minimum acquaintance with basic set theory, the barest minimum about complex numbers, and the beautiful Euler formula. The book explains almost everything beyond this that one would need to know, but does occasionally use technical terms which are not explicitly defined. Also, the glossary is good but not great, and the index is lacking some key terms. Also at times the author could be more clear in stating how one knows something (i.e., by citing where in a previous chapter it was introduced) instead of assuming that one's recall of the previous chapters is perfect.

These relatively minor cavils aside, I think the book is great. In addition to explaining why there are no general formulas for solving by algebraic means polynomials in rational coefficients of fifth degree and higher, it also explains why one can't square the circle, and why only certain polygons are constructible by ruler and compass, classic conundrums. It does this through the use of abstract algebra, i.e., fields, field extensions, and groups. If you stick with the line of argument, you will be impressed by the power and logic of higher mathematics, and the ability of seemingly abstruse and ethereal concepts to solve concrete problems.

So anyway, the answer to why the quintic polynomial with rational coefficients is not in general solvable is found in chapter 10, section 1. I.e., you will need to plow through 8 chapters to understand it (not 9 chapters, because chapter 4 is a digression into why the circle can't be squared). By Theorem 10.4, the Galois group of a monic irreducible polynomial of prime degree p (e.g., a quintic) with exactly 2 non-real complex roots is the Sp symmetry group (i.e., in the case of the quintic, the S5 symmetry group). In Chapter 9 it was shown that the Sn group for n greater than or equal to 5 is not soluble. Therefore the Galois group of a quintic such as x^5 - 8x + 2, which has exactly two non-real complex roots, being the S5 symmetry group, is nonsoluble. Therefore the roots of this polynomial cannot be calculated from its coefficients (1, -8, and 2) by any combination of addition, subtraction, multiplication, division, and taking of roots (any degree root, such as square root, cube root, eleventh root, you name it). This doesn't mean that there aren't some quintics where this can be done, but by creating even a single counterexample, we've proven that there is no general formula applicable to all quintics.

Now for a polynomial of any degree n higher than five one can always create an example of an insoluble polynomial by multiplying an insoluble quintic by another polynomial of degree n-5 as a factor (e.g., an insoluble sixth-degree polynomial can be created from the product of an insoluble quintic and (x-2), an insoluble seventh-degree polynomial can be created from the product of an insoluble qunitic and (x^2-1), etc.), so having an insoluble factor it is also insoluble. Thus there is no general formula for any polynomial of degree higher than 5, either.

If you're like me it will take you about 2 months working about an hour a day to get to this point. Is it worth it? Ask Galois.

I needed a book with examples of normal extensions. This one was very helpful.