Fields and Rings (Chicago Lectures in Mathematics Series)

Rating:
Summary: Pretty good
Review: This book is an advanced treatment of field theory and Galois theory and is meant for those readers who have a substantial background in graduate algebra. The subject matter used to be thought of as purely mathematical, but due to the influence of the field of cryptography, it now has many applications. I only read part 1 of the book, so my review will be confined to this part.

The author begins the discussion with field extensions. One can view a field L containing another field K as a vector space over K, and the dimension of L (as a vector space) is then called the 'dimension' of L over K. If one considers a subfield K of a field M, and an additional element u in M, then there is a smallest subfield of M containing K and u. Calling this field K(u), u can be either transcendental or algebraic over K. The author then proves some elementary properties of the field K(u), showing the existence of an irreducible polynomial for u over K. This then motivates him to call a field L containing K 'algebraic' over K if every element of L is algebraic over K. Otherwise L is called 'transcendental' over K. The dimension of K(u) over K is called the degree of u over K. Finding the degree of u can be done by finding the irreducible polynomial for u. The author also proves the arithmetic relation between the dimensions of towers of fields, and this allows him to prove the famous results on the impossibility of ruler and compass constructions. For a field L that lies between fields K and M the author studies the 'stability' of L over K, meaning that every automorphism of M/K sends L into itself. The correspondence between stable fields and normal subgroups of the Galois group of M/K is proven. Splitting fields are introduced as devices to obtain fields that are normal over a given field. A criterion for a splitting field that does not involve polynomials is proven, and the author gives tools that deal with fields of non-zero characteristic, these tools motivating the definition of separability. Splitting fields are normal in characteristic 0, but one must add separability for the same to hold in characteristic p. The unsolvability of the quintic is shown via a discussion on radical extensions of fields. For a field K of characteristic 0, and for a field L lying between K and another field M, where M is a radical extension of K, the author proves in detail that the Galois group of L/K will be solvable. Then if one has a polynomial with coefficients in K, then the Galois group of this polynomial is defined to be the Galois group of a splitting field of the polynomial over K. The Galois group of the polynomial is thought of as a group of permutations of the roots of the polynomial. The author then proves that if K has characteristic 0 and L is a radical extension of K which contains a root of the polynomial, then the Galois group of the polynomial over K is solvable.

Those readers involved in cryptography will find a discussion of finite fields in Part 1. The author's goal is to find the finite fields and determine their structure. He first proves that every nth power of a prime number p will yield a field with p^n elements. The author shows that the Galois theory of finite fields is simple by proving that if K is a finite field contained in another finite field L, then L is normal over K and the Galois group of L/K is cyclic.

The author also shows how the Galois group of an equation can be found explicitly for the cubic and quartic equations. He shows first that for the Galois group of a separable irreducible cubic over a field K is either the alternating group A(3) or the symmetric group S(3). If the characteristic of K is not equal to 2, then it is A(3) if and only if the discriminant is a square in K. For a separable irreducible quartic over K, then for the degree over K of the splitting field of the resolvent cubic of this polynomial, the Galois group is S(4) if the degree is 6, A(4) if the degree is 3, V (a particular normal subgroup of S(4)) if the degree is 1, and either the group of order 8 or cyclic of order 4 if the degree is 2.

Also in part 1, the author studies the reducibility of an equation of the form x^n -a over an arbitrary field. He addresses this reducibility by first proving that one only need be concerned for the case where n is a prime power. Then if p is prime, and "a " does not have any pth root in the field K, then if the prime is odd, then the equation is irreducible over K for any n. If p = 2 and the characteristic of K is 2, then the equation is irreducible over K for any n. If p = 2, n is greater than or equal to 2, and the characteristic of K is not 2, then the equation is irreducible over K if and only if -4a is not a fourth power in K. The author also proves the fundamental theorem of algebra using Galois theory. He does this by first showing that if every extension of K has degree divisible by a prime p, then every extension of K has degree a power of p.