x Preface to Second Edition Noncommutative rings are now discussed earlier, so that left and right modules can appear at the same time. The existence of free groups is shown more simply (using only deletions instead of deletions and insertions). In fact, much of the section on presentations has been rewritten. Division rings and Brauer groups are introduced in the same chapter as the Wedderburn–Artin Theorems their discussion then continues after cohomology has been studied. The section on Grothendieck groups has been completely rewritten. Some other items appearing here that were not treated in the first edition are Galois Theory for infinite extensions, the Normal Basis Theorem, abelian categories, and module categories. The Table of Contents enumerates the highlights of each chapter here are more details. Chapters 1 and 2 present the elements of Group Theory and of Commutative Algebra, along with Linear Algebra over arbitrary fields. Chapter 3 discusses Galois Theory for finite field extensions (Galois Theory for infinite field extensions is discussed in Chapter 6, after inverse limits have been introduced). We prove the insolvability of the general polynomial of degree 5 and the Fundamental Theorem of Galois Theory. Among the applications are the Fun- damental Theorem of Algebra and Galois’s Theorem that a polynomial over a field of characteristic 0 is solvable by radicals if and only if its Galois group is a solvable group. The chapter ends by showing how to compute Galois groups of polynomials of degree ≤ 4. Chapter 4 continues the study of groups, beginning with the Basis Theorem and the Fundamental Theorem for finite abelian groups (finitely generated abelian groups are discussed in Chapter 8). We then prove the Sylow Theorems (which gen- eralize the Primary Decomposition to nonabelian groups), discuss solvable groups, simplicity of the linear groups PSL(2,k), unitriangular groups, free groups, presen- tations, and the Nielsen–Schreier Theorem (subgroups of free groups are free). Chapter 5 continues the study of commutative rings, with an eye to discussing polynomial rings in several variables unique factorization domains Hilbert’s Ba- sis Theorem applications of Zorn’s Lemma (including existence and uniqueness of algebraic closures, transcendence bases, and inseparability) L¨ uroth’s Theorem aﬃne varieties Nullstellensatz over C (the full Nullstellensatz, for varieties over ar- bitrary algebraically closed fields, is proved in Chapter 10) primary decomposition of ideals the Division Algorithm for polynomials in several variables Buchberger’s algorithm and Gr¨ obner bases. Chapter 6 introduces noncommutative rings, left and right R-modules cate- gories, functors, natural transformations, and categorical constructions free mod- ules, projectives, and injectives tensor products adjoint functors flat modules inverse and direct limits infinite Galois Theory. Chapter 7 continues the study of noncommutative rings, aiming toward Repre- sentation Theory of finite groups: chain conditions Wedderburn’s Theorem on finite division rings Jacobson radical Wedderburn–Artin Theorems classifying semisimple rings. These results are applied, using character theory, to prove Burn- side’s Theorem (finite groups of order pmqn are solvable). After discussing multiply transitive groups, we prove Frobenius’s Theorem that Frobenius kernels are nor- mal subgroups of Frobenius groups. Since division rings have arisen naturally, we

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