Abstract Algebra Theory and Applications - Computer Science ...

416 INDEXEquivalence relation, 15Euclidean algorithm, 29Euclidean domain, 286Euclidean group, 178Euclidean inner product, 175Euclidean valuation, 286Euler φ-function, 94Euler, Leonhard, 94, 342Extensionalgebraic, 325field, 322finite, 328normal, 372radical, 378separable, 348, 369simple, 325External direct product, 144Faltings, Gerd, 342Feit, W., 161, 217Fermat’s factorization algorithm, 105Fermat’s Little Theorem, 94Fermat, Pierre de, 94, 342Ferrari, Ludovico, 271Ferro, Scipione del, 270Field, 233algebraically closed, 332base, 322conjugate, 384extension, 322fixed, 371Galois, 349of fractions, 280of quotients, 280prime, 292splitting, 334Finitely generated group, 191Fior, Antonio, 270First Isomorphism Theoremfor groups, 162for rings, 242Fixed point set, 205Freshman’s Dream, 347Frobenius map, 361Functionbijective, 10Boolean, 214, 311composition of, 10definition of, 9domain of, 9identity, 12injective, 10invertible, 13one-to-one, 10onto, 10order-preserving, 310range of, 9surjective, 10switching, 214, 311Fundamental Theoremof Algebra, 333, 382of Arithmetic, 30of Finite Abelian Groups, 193of Galois Theory, 373Gödel, Kurt, 308Galois field, 349Galois group, 365Galois, Évariste, 46, 377Gauss’s Lemma, 288Gauss, Karl Friedrich, 290Gaussian integers, 237Generator of a cyclic subgroup, 57Generators for a group, 191Glide reflection, 179Gorenstein, Daniel, 161Greatest common divisorof elements in a UFD, 292of two integers, 27of two polynomials, 263Greatest lower bound, 296Greiss, R., 161Grothendieck, A., 342Groupp-group, 192, 220abelian, 41action, 203alternating, 79automorphism of, 150center of, 87, 168, 207