vi Contents 3.1.1. Classical Formulas and Solvability by Radicals 181 3.1.2. Translation into Group Theory 184 3.2. Fundamental Theorem of Galois Theory 192 3.3. Calculations of Galois Groups 212 Chapter 4. Groups II 223 4.1. Finite Abelian Groups 223 4.1.1. Direct Sums 223 4.1.2. Basis Theorem 230 4.1.3. Fundamental Theorem 236 4.2. Sylow Theorems 243 4.3. Solvable Groups 252 4.4. Projective Unimodular Groups 263 4.5. Free Groups and Presentations 270 4.6. Nielsen–Schreier Theorem 285 Chapter 5. Commutative Rings II 295 5.1. Prime Ideals and Maximal Ideals 295 5.2. Unique Factorization Domains 302 5.3. Noetherian Rings 312 5.4. Zorn’s Lemma and Applications 316 5.4.1. Zorn’s Lemma 317 5.4.2. Vector Spaces 321 5.4.3. Algebraic Closure 325 5.4.4. uroth’s Theorem 331 5.4.5. Transcendence 335 5.4.6. Separability 342 5.5. Varieties 348 5.5.1. Varieties and Ideals 349 5.5.2. Nullstellensatz 354 5.5.3. Irreducible Varieties 358 5.5.4. Primary Decomposition 361 5.6. Algorithms in k[x1,...,xn] 369 5.6.1. Monomial Orders 370 5.6.2. Division Algorithm 376 5.7. Gr¨ obner Bases 379 5.7.1. Buchberger’s Algorithm 381 Chapter 6. Rings 391 6.1. Modules 391 6.2. Categories 418 6.3. Functors 437 6.4. Free and Projective Modules 450
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