The second edition of Roman’s “Field Theory” (for the review of the first ed. (1995) see Zbl 0816.12001) offers a graduate course on Galois theory. The first chapter collects, without proofs, preliminary results from algebra, ranging from unique factorization and Euclidean domains to the theorems on Sylow subgroups and tensor products. The remaining chapters are divided into three parts, the first of which discusses field extensions and introduces concepts such as splitting fields, algebraic closure, separability, algebraic independence and transcendental extensions. Part II presents Galois theory, and starts with a historical perspective of Galois theory (viewed as the problem of solving polynomial equations by radicals – this chapter is not contained in the first edition) from the Babylonians to Galois. Included are a detailed discussion of the Galois groups of cubic and quartic polynomials, a proof of the fundamental theorem of algebra, the theory of finite fields, cyclotomic extensions, and Wedderburn’s theorem. The last part deals with Kummer theory and the Galois group of extensions generated by radicals.
The author’s approach is mainly standard; an exception is his description of the Galois correspondence, which is more abstract than usual, and introduces a variety of new notions like closed elements, top and bottom elements, closure points etc. Perhaps a few words on the merits of such an approach would have been helpful for readers who already know some Galois theory, or for instructors who have to pick a textbook. I would also like to remark that \(\mathbb Z_p\) is not a particularly good notation for the residue class group modulo \(p\).
The clarity of exposition and lots of exercises make this a suitable textbook for a graduate course on Galois theory.