Bibliography

This is primarily a list of books where the topics are pursued further (and which were often used as sources); it is followed by a list of papers referred to in the text as well as a small selection of articles having a bearing on the text. FA refers to the sequel of this book: Further Algebra and Applications. See below.

Anderson, F. W. and Fuller, K. R. (1973) Rings and Categories of Modules, Graduate Texts in Mathematics 13, Springer Verlag, Berlin. Artin, E. (1948) Galois Theory, Notre Dame Math. Lectures No.2, Notre Dame, IN. Artin, E. (1957) Geometric Algebra, Interscience, New York. Barwise, J. (ed.) (1977) Handbook of Logic, North-Holland, Amsterdam. Birkhoff, G. (1967) Lattice Theory (3rd edn), AMS, Providence, RI. Bourbaki, N. (1961-80) Algebre, Chs. 1-10, Hermann, Paris, later Masson, Paris. Bourbaki, N. (1984) Elements d'Histoire de Mathematiques, Masson, Paris. Burnside, W. (1911) Theory of Groups of Finite Order (2nd edn), Cambridge Univer- sity Press; reprinted 1955, Dover, New York. Chase, S. U., Harrison, D. K. and Rosenberg, A. (1965) Galois Theory and Cohomol• ogy Theory of Commutative Rings, Mem. Amer. Math. Soc. 52, AMS, Providence, RI. Chevalley, C. (1951) Introduction to the Theory ofAlgebraic Functions of One Variable, No.24, AMS Colloquium Publications, Providence, RI. Cohen, P. J. (1966) Set Theory and the Continuum Hypothesis, Benjamin, New York. Cohn, P. M. (1981) Universal Algebra (2nd edn), Reidel, Dordrecht. Cohn, P. M. (1985) Free Rings and Their Relations (2nd edn), LMS Monographs No.19, Academic Press, New York. Cohn, P. M. (1991) Algebraic Numbers and Algebraic Functions, Chapman & Hall! CRC Press. Cohn, P. M. (1995) Skew Fields, Theory of General Division Rings, Encyclopedia of Mathematics and its Applications, Vol. 57, Cambridge University Press. Cohn, P. M. (2000) Introduction to Ring Theory, SUMS, Springer Verlag, London. Cohn, P. M. (2003) Further Algebra and Applications, Springer Verlag, London, referred to as FA. Dedekind, R. (1894) Uber die Theorie der ganzen algebraischen Zahlen, XI. Supple• ment zu Dirichlets Vorlesungen tiber Zahlentheorie, 2. Aufl.; reprinted 1964, Vieweg, Braunschweig.

449 450 Basic Algebra

Endler, O. (1972) Valuation Theory, Springer Verlag, Berlin. Fossum, R. M. (1973) The Divisor Class Group of a Krull Domain, Springer Verlag, Berlin. Fuchs, L. (1970, 1973) Abelian Groups I, II, Academic Press, New York. Galois, E. (1951) Oeuvres Mathematiques, Gauthier-Villars, Paris. Hall, M. Jr. (1959) The Theory of Groups, Macmillan, New York. Hartshorne, R. (1977) Algebraic Geometry, Graduate Texts in Math. 52, Springer Verlag, Heidelberg. Hilbert, D. (1897) Bericht uber die Theorie der algebraischen Zahlkorper, lahrber. DMV iv; reprinted in Vol. 1 of the Collected Works. Huppert, B. (1967) Endliche Gruppen I, Grundl. d. math. Wiss. 134, Springer Verlag, Berlin. Jacobson, N. (1985, 1989) Basic Algebra (2nd edn), I, II, W. H. Freeman, New York. Kaplansky, I. (1972) Set Theory and Metric Spaces, Allyn & Bacon, Boston. Klein, F. (1884) Lectures on the Icosahedron; reprinted 1956, Dover, New York. Lam, T. Y. (1980) The Algebraic Theory of Quadratic Forms, Adv. Book Progr., Benjamin/Cummings, Reading, MA. Lang. S. (1970) Algebraic Number Theory, Addison-Wesley, Reading, MA. Lang, S. (1984) Algebra (2nd edn), Addison-Wesley, Reading, MA. Lang, S. (2002) Algebra (revised 3rd edn) Springer Verlag, Berlin. Lidl, R. and Pilz, G. (1984) Applied Abstract Algebra, Springer Verlag, Berlin. Mac Lane, S. (1971) Categories for the Working Mathematician, Springer Verlag, Berlin. Mahler, K. (1981) p-adic Numbers and their Functions (2nd edn), Cambridge Univer- sity Press. Matsumura, H. (1985) Commutative Rings, Cambridge University Press. Nagata, M. (1962) Local Rings, Interscience, New York. Neukirch, J. (1986) Class Field Theory, Grundl. d. math. Wiss. 280, Springer Verlag, Heidelberg. Ore, O. (1953) Cardano, the Gambling Scholar, Princeton University Press, Prince- ton, NJ. Rotman, J. J. (1965) The Theory of Groups, An Introduction, Allyn & Bacon, Boston. Rowen, L. H. (1988) Ring Theory I, II, Academic Press, New York. Rudin, W. (1966) Real and Complex Analysis, McGraw-Hill, New York. Scharlau, W. (1985) Quadratic and Hermitian Forms, Grundl. d. math. Wiss. 270, Springer Verlag, Heidelberg. Semple, J. G. and Roth, L. (1949) Introduction to Algebraic Geometry; reprinted 1987, Clarendon Press, Oxford. Serre, J.-P. (1979) Local Fields, Graduate Texts in Math. 67, Springer Verlag, Heidelberg. Sierpinski, W. (1956) Cardinal and Ordinal Numbers, Pan. Wyd. Nauk, Warsaw. van der Waerden, B. L. (1971, 1976) Algebra I, II, Springer Verlag, Berlin. Weber, H. (1894, 1896, 1908) Lehrbuch der Algebra I-III, Teubner, Leipzig; reprinted 1963, Chelsea, New York. Welsh, D. J. A. (1976) Matroid Theory, LMS Monographs 8, Academic Press, London. Bibliography 451

White, N. (ed.) (1986) Theory of Matroids, Encyclopedia of Mathematics and its Applications, Vol. 26, Cambridge University Press.

List of Papers

Bass, H. [1960] Finitistic dimension and a homological generalization of semi• primary rings, Trans. Amer. Math. Soc. 95, pp. 466-488. Cohn, P. M. [1966] Some remarks on the invariant basis property, Topology 5, pp.2l5-228. Cohn, P. M. [ 1973] Unique factorization domains, Amer. Math. Monthly 80, pp. 1-17. Cohn, P. M. [1997] Cyclic Artinian modules without a composition series, f. London Math. Soc. (2) 55, pp. 231-235. Deligne, P. R. [1973] Varietes unirationnelles non rationnelles, Sem. Bourbaki 1971I2, Exp. 402, Lecture Notes in Math. 317, Springer Verlag, Heidelberg. Eilenberg, S. and Mac Lane, S. [1945] General theory of natural equivalences, Trans. Amer. Math. Soc. 58, pp. 231-294. Hartley, B. [1977] Uncountable Artinian modules and uncountable soluble groups satisfying Min-n, Proc. London Math. Soc. (3) 35, pp. 55-75. Hodges, W. A. [1974] Six impossible rings, f. Algebra 31, pp. 2l8-244. Kaplansky, I. [1958] Projective modules, Ann. Math. 68, pp. 372-377. Lenstra Jr., H. W. [1974] Rational functions invariant under a finite , Invent. Math. 25, pp. 299-325. Nagata, M. [1957] A remark on the unique factorization theorem, f. Math. Soc. Japan 9, pp. 143-145. Pierce, R. S. [1967] Modules over commutative regular rings, Memoirs of the AMS No.70, AMS, Providence, RI. Rota, G.-c. [1964] On the foundations of combinatorial theory I. Mobius functions, Z. Wahrsch. 2, pp. 340-368. Schur, I. [1905] Neue Begriindung der Theorie der Gruppencharaktere, Sitzungsber. d. Preuss. Akad. d. Wiss., pp. 406-432. Steinitz, E. [1910] Algebraische Theorie der Korper, J. Reine Angew. Math. 137, pp. 167-309; reprinted 1930, Teubner, Leipzig, 1950, Chelsea, New York. Steinitz, E. [1911, 1912] Rechteckige Systeme und Moduln in algebraischen Zahl• korpern I, II, Math. Ann. 7l, pp. 328-354, 72, pp. 297-345. Swan, R. G. [1969] Invariant rational functions and a problem of Steenrod, Invent. Math. 7, pp. 148-158. Voskresenskii, V. E. [1973] Fields of invariants of abelian groups, Uspekhi Mat. Nauk SSSR 28, pp. 77-102 (in Russian). Witt, E. [1931] Uber die Kommutativitat endlicher Schiefkorper, Hamb. Abh. 8, p.413. List of Notations

In some cases a page number is given where the term is first used or defined.

Number Systems

N the natural numbers No the natural numbers with 0 Z the integers Q the rational numbers Q+ the non-negative rational numbers R the real numbers C the complex numbers Urn the group of m-th roots of unity Z(pOO) the group of all pn_th roots of 1, for n = 1,2, ... Z/(n) or Zin the integers mod n 27 U(n) the group of units mod n 221 Fq the field of q elements 224 Zp the p-adic integers 315 Qp = Zp[p-lj the p-adic numbers 315

Set Theory o the empty set xi, 1 IXI cardinal of the set X 2 Y'(X) power set (set of all subsets) of X 6 x\Y complement of Y in X xi yX set of all mappings from X to Y 5 ~o aleph-null, the cardinal of N 2

453 454 Basic Algebra Number Theory max(a, b) the larger of a, b min (a, b) the smaller of a, b alb a divides b (a, b) highest common factor (HCF) of a and b [a, b] least common multiple (LCM) of a and b 8ij Kronecker delta xi /-L(n) Mobius function 158 f/J(m) Euler function 161 m(x) cyclotomic polynomial 219

Group Theory

Symn symmetric group of degree n 32 Altn alternating group of degree n 33 sgn a sign of the permutation a 33 Cn of order n 27 Dm dihedral group of order 2m 26 G' derived group of G 39 N

Rings and Modules mvn space of all m x n matrices over V 97 mv space of m-component columns over V (= mvl) 97 vn space of n-component rows over V (= I vn) 97 9J1n(R) or Rn n x n matrix ring over R 97 Lat(M) lattice of all submodules of M 89 Hom(U, V) set of all homomorphisms from U to V 83 End(U) ring of all endomorphisms of U 83 U®V tensor product of U and V 117 tM torsion submodule of M 90 RO opposite of the ring R 82 RX set of non-zero elements in R 80 Al augmented algebra of A 132 Ann (X) annihilator of X 84 Ass(M) assassinator of M 380 List of Notations 455

Supp(M) support of M 358 Rs or Rp localization of R at 5 (or at the complement of p) 354f Ja radical of an ideal a 353 K[x] polynomial ring on x over K 166 K[[xll formal power series ring on x over K K(X} free K-algebra on X 134 sModR category of (5, R)-bimodules 86 $'(R) field of fractions of commutative integral domain R 428 TIM; direct product of modules 87 UM; direct sum (coproduct) of modules 87 'In(R) upper triangular matrices over R 133

Field Theory

[V: k] dimension of the k-space V 190 k(ex) field generated by ex over k 191 k[ex] ring generated by ex over k 191 Gal(EjF) group of the Galois extension EjF 211 T(x) trace of x 153 N(x) norm of x 153 U1-V orthogonal sum of U and V 252f Ul. orthogonal complement of U 251 (aJ, ... , an) quadratic form (in diagonal form) 254

Categories (mappings resp. homomorphisms are understood)

Ens sets 65 Gp groups 65 Ab abelian groups 66 Rg rings 65 Top topological spaces 65 Mod modules 65 vec vector spaces 68 col column vectors 68 Fun(I, d) functors from I to .91 69 Author Index

Abel, Niels Henrik (1802-29) 238 Euler, Leonhard (1707-83) 17, 161,242, 247f. Abhyankar, Shreeram S. (1931-) 445 Faltings, Gerd (1954-) 362 Adyan, Sergei 1. 28 Fermat Pierre de (1601-65) 242, 362, 371 Akgiil, M. II 0 Ferrari, Lodovico (1522-65) 238 Amitsur, Shimshon A. (1921-94) 146 Ferro, Scipio del (1465-1526) 238 Arf, Cahit (1910-) 303 Fitting, Hans (1906-38) 48 Artin, Emil (1898-1962) 89, 139, 209, 216, 279, Flanders, Harley (1925-) 425 283,285,316,319,432 Franke, E. 186 Frattini, Giovanni (1852-1925) 44, 46f. Baer, Reinhold (1902-79) 116, 130 Frobenius, F. Georg (1849-1917) 203,224 Banach, Stefan (1892-1945) 321 Bass, Hyman (1932-) 144 Galois, Evariste (1811-32) 206, 2I1ff., 216, 238f., Becker, Eberhard 285 242, 244f. Bernstein, Felix (1878-1956) 4,77 Gauss, Carl F. (1777-1855) 134,217,242,248, Binet, Jacques P. M. (1786-1856) 182 338 Bolzano, Bernard (1781-1848) 2 Gelfand, Izrael M. (1913-) 321 Boole, George (181H4) 70ff. Godel, Kurt (1906-78) 7, 10 Bourbaki, Nicolas (1901-) 380 Golod, Evgenii S. (1935-) 176ff., 179 Brauer, Richard D. (1901-77) 152 Goodearl, Kenneth R. (1945-) 129 Burali-Forti, Cesare (1861-1931) 6 Gottschalk, Walter H. 78 Burnside, William (1852-1927) 28, 178 Grassmann, Hermann G. (1809-77) 184 Gregory, James (1638-75) 238 Cantor, Georg F.L.P. (1845-1918) 2,4, 7f., 14, Grothendieck, Alexander (1928-) 291 275 Capelli, Alfredo (1858-1916) 247 Hahn, Hans (1879-1934) 321 Cardano, Girolamo (1501-76) 238 Hall, Philip (1904-82) 18,44, 49 Cartan, Henri P. (1904-) 258 Halmos, Paul R.(1914-) 19 Castelnuovo, Guido (1865-1952) 408 Hamel, Georg (1877-1954) 401 Cauchon, Gerard 446 Hamilton, Sir William R.(l805-{)5) 148, 299 Cauchy, Augustin Louis (1789-1857) 182, 275f. Harriot, Thomas (1560-1621) 287 Cayley, Arthur (1821-95) 32, 135 Harrison, David K. (1931-90) 297 Chevalley, Claude C (1909-84) 227, 333 Hartley, Brian (1939-94) 146 Clifford, William K. (1845-79) 260,265 Hasse, Helmut (1898-1980) 296 Cohen, Irving S. (1917-55) 395 Hausdorff, Felix (1868-1942) 22,435 Cohen, Paul J. (1934-) 7, 10 Hensel, Kurt (1861-1941) 311, 322, 340f. Cohn, Paul M. 108, 146, 185 Hilbert, David (1863-1941) 174f., 221, 347, 361, 392 De Morgan, Augustus (1806-71) 70 Hodges, Wilfrid A. (1941-) 60 Dedekind, J.w.Richard (1831-1916) 2, 55, 115, Holder, Otto (1859-1937) 36 206,209,216,275,351, 362f., 365, 395 Hopkins, Charles (1902-39) 139, 145f. Deligne, Pierre Rene (1945-) 408 Huntingdon, Edward V. (1874-1952) 78 Descartes, Rene (1596-1650) 287 Dieudonne, Jean A. (1906-92) 137, 258 Ingleton, Aubrey W. (1921-2000) 445 Dilworth, Robert P. (1914-93) 18 Iversen, Birger 371 Diophantos (~250) 238 Dirichlet, Peter Gustav Lejeune Jacobi, Carl Gustav Jacob (1804-51) 43 (1805-59) 2, 216, 222, 370 Jacobson, Nathan (1910-99) 142f. Jordan, Camille (1838-1922) 36 Eilenberg, Samuel (1913-98) 69 Eisenstein, Ferdinand Gotthold M. (1823-52) Kaplansky, Irving (1917-) 376 199f. Klein, Avraham A. 146 Erdos, Paul (1913-96) 21 Klein, C. Felix (1849-1925) 26, 185

457 458 Basic Algebra

Knebusch, Martin 285 Ruffini, Paolo (1765-1822) 238 Konig, Denes (1884-1944) 21 Russell, (Lord) Bertrand A. W. (1872-1970) 1, Konig, Gyula (Julius) (l849-1913) 5, 246 10 Krasner, Mark A. (l920-97) 345 Kronecker, Leopold (l823-9l) 195, 221, 246 Sarges, Heidrun 361 Krull, Wolfgang (1899-1971) 90, 96, 395, 434 Scharlau, Winfried 283 Kummer, Ernst-Eduard (1810-93) 347, 351, Schmidt, Otto Yu. (l891-1956) 96 362f., 441 Schmidt, Friedrich Karl (l901-77) 425 Kuratowski, Kazimierz (1896-1980) 10 Schreier, Otto (l901-29) 37, 56, 279 Kurosh, Aleksandr G. (1908-71) 178 Schroder, F.W.K.Ernst (1841-1902) 4, 77 Schur, Issai (l875-1941) 137, 139,220 Lagrange, Joseph L. (l736-1813) 28, 104,237 Seidenberg, Abraham(1916-) 389 Laplace, Pierre S. Marquis de (l749-1827) 154, Serre, Jean-Pierre (1926-) 174 184 Shafarevich, Igor R. (l923-) 176£. Lasker, Emanuel (l868-194l) 380 Sheffer, H. M. (l883-1964) 78 Laurent, Pierre Alphonse (1803-54) 133, 166 Sierpinski, Wad'aw (l882-1969) 22 Legendre, Adrien-Marie (l752-1833) 247f., 290 Skolem, A. Thoralf (1887-1963) 265 Leibniz, Gottfried Wilhelm, Freiherr von Speiser, Andreas (l885-1970) 438 (1646-1716) 173 Sperner, Emanuel (l905-80) 22 Leicht, Johann B. 297 Spitzlay, K.-E. 285 Lenstra Jr., Hendrik W. 404 Steinitz, Ernst (l871-1928) 228, 238, 374, 432, Levitzki, Jacob (l904-56) 139, 146 435 Lindemann, C. L. Ferdinand von (1852-1939) Stone, Marshall H. (l903-89) 76 194 Sturm, Jacques-Charles-Frans:ois (1803-55) Liouville, Joseph (l809-82) 322 288ff. Lorenz, Falko 297 Swan, Richard G. (1933-) 404 Lubell, David 22 Sylow, P. Ludvig M. (l832-1918) 37 Liiroth, Jakob (1844-1910) 407 Sylvester, James Joseph (l814-97) 186, 285f. Szekeres, George (l911-) 21 Mac Lane, Saunders (1909-) 69, 423£, 445 Mahler, Kurt (1903-88) 328 Tarski, Alfred (1902-83) 22 Maschke, Heinrich (1853-1908) 162 Tartaglia, Niccolo (1500-57) 238 Mazur, Stanislaw (l905-) 321 Merkuryev, A. 305 Vahlen, Karl Theodor (1869-1945) 247 Moebius, August Ferdinand Vandermonde, Alexandre Theophile (l735-96) (1790-1868) 158f. 232 Moore, Eliakim Hastings (l862-1932) 224 Viete, Frans:ois (l540-1603) 238 Morita, Kiiti (l915-95) 100 Vinberg, Ernest B. 177 Mumford, David B. (l937-) 311 Voskresenskii, Valentin E. 404

Nagata, Masayoshi (l927-) 185, 395 Wall, Charles Terence Clegg (l936-) 296 Nakayama, Tadasi (1912-64) 144, 395 Wantzel, Pierre L. (l814-48) 194 Newton, Sir Isaac (1642-1727) 325 Warning, Ewald 227 Noether, A. Emmy (l882-1935) 61, 89, 139,265, Waterhouse, William C. 446 365£, 380, 391, 409, 438 Weber, Heinrich (1842-1913) 221 Novikov, Petr S. 28 Wedderburn Joseph H. Maclagan (l882-1948) 13 7ff., 156, 226 Ore, Oystein (1899-1968) 59 Weisner, Louis (l899-1988) 162 Ornstein, Donald S. 45 Whaples, George (l914-8l) 316 Ostrowski, Alexander (l893-1986) 319f. Whitney, Hassler (1907-89) 445 Wielandt, Helmut W. (1910-2001) 45,47 Perlis, Sam (l913-) 142 Wiles, Andrew (l953-) 362 Pfister, Albrecht (l934-) 305 Witt, Ernst (1911-91) 43, 226, 268ff., 291 ff. Pierce, Richard S. (l927-92) 104 Pliicker, Julius (1801-68) 185 Yoneda, Nobuo 78 Poincare, J. Henri (l854-1912) 32, 174 Zafrullah, Muhammad (l942-) 394 Rabinowitsch, J. L. 393 Zariski, Oscar (l899-1986) 380, 393, 408 Rados, G. 246 Zassenhaus, Hans J. (l912-91) 37,56 Ramsey, Frank Plumpton (1903-30) 19f. Zech, Theodor 224 Riemann, Bernhard (l826-66) 203, 309 Zelmanov, Efim I. (l955-) 28 Roos, Jan-Erik. 96 Zermelo, Ernst F.F. (l871-1953) 10 Rota, Gian-Carlo (l932-99) 157 Zorn, Max A. (1906-93) 10 Subject Index

Generally, non-X, un-X, in-X is listed under X. abelian group 25 augmentation ideal 132, 167, 293 abelianization 67 augmented algebra 132 absolute value 273, 312 automorphism 27, 80, 211 absorptive law 53 axiom of choice 10,60 acquaintanceship graph 16 acyclic 17 Baer's criterion 116 addition (mod 2) 81 balanced mapping 122 additive function 173 basis 105, 398 - functor II 0 - theorem for abelian groups 38 adjacency matrix 23 bidual 126 adjoint associativity 119, 123 bifunctor 87 affine group 227, 244 bilinear form 249ff. aleph 2 - mapping 117, 122 algebra 131 bimodule 83 algebraic element 192 binary form 249 - equation 376 Binet-Cauchy identity 182 - extension 193, 403 Boolean algebra 70, 134 - integer 134 - polynomial 71 - set 378 - ring 80 algebraically closed 201, 285, 431 Brauer group 152, 296 - dependent 402 Burnside problem 28, 178 alternating form 256, 298, 301 - matrix 298 cancellation 31 - group 33 cardinal (number) Iff. anisotropic part 257, 270, 293 Cartan-Dieudonne theorem 258 annihilator 84 Castelnuovo-Zariski theorem 408 anti-chain 8, 63f. casus irreducibilis 247 anticommutative 166f. category 65 antiderivation 180 Cauchy sequence 275, 314 antihomomorphism 66, 83 Cayley's theorem 32, 135, 214 approximation theorem 316, 369 central chain 39 archimedean absolute value 313 - simple algebra 150 - ordering 277 centralizer 30, 149 Arf invariant 303 centrally primitive idempotent 103 arrow 17 centre 30, 131 Artin's theorem 209 chain 8 Artin-Schreier extension 444 - condition 60 Artin-Schreier theory 279ff. character (group) 125 Artinian module, ring 89 characteristic of field 189 assassinator 380 -- prime ideal 297 associated elements 349 -- ring 80, 104 - prime ideal 380 - function 6 associative law 25, 53 - polynomial 153, 175 atom 75, 193, 349 - 46 atomic Boolean algebra 78 Chevalley's lemma 333 - domain 350 chief factor, series 36

459 460 Basic Algebra

Chinese remainder theorem 102, 115 -- rational function 405 class 65 -- symmetric group 26 - equation 30 Delian problem 194 - of nilpotence 40 denominator 335, 354f. classical logic 71 Clifford algebra 260 dense embedding 275, 393 - group 265f. - functor 67 closed set 378ff. density principle 228 cofinal 14 dependence relation 397ff. cofinite subset 70 derivation 171, 415ff. 1. S. Cohen's theorem 395 derivative 204 coimage, cokernel 85 derived group, series 39 comaximal ideals 102 determinant 182 comma category 69 - of a form 252 commutative diagram 85 diagonal argument 8 -law 25,53 diamond lattice 57 - ring 79 dicyclic group 50 commutator 39, 42 different 395 companion matrix 155 digraph 17 complement 56, 92 dihedral group 26, 50 complementary graph 16 Dilworth's theorem 18 complete lattice 55 dimension 140, 249, 387, 403 - ordered field 275 - index 295 - space 314 direct power 87 completely primary ring 104, 187, 386 - product 27, 40, 87 completion 277, 315 -- of rings 101 composite of fields 428 - sum of modules 87 composition series 36 directed graph 52 compound matrix 182 - system 64, 202 concrete category 66 Dirichlet box principle 2 conductor 391 Dirichlet's theorem 222 cone 279f. discrete absolute value 313 congruent matrices 250 - rank 1 valuation 308 conjugacy class 30 discriminant 231, 263 conjugate 30, 42, 199,213 disjunctive normal form 72 conjunctive normal form 72 distributive lattice 59, 96 connected graph 19 - law 59, 79, 119 conorm mapping 370 divisible module 116 consistent system 377 division algebra 133, 150 constant 171,415 - ring 80 continuum hypothesis 7 dominate 332 contracted ideal 356 dual basis lemma 114 converge 275 - categories 67 co(ntra)variant functor 66 - group 125 coordinate ring 377 - homomorphism 71 core of a field 280 duality 67f. coset (space) 27ff. countable 2 edge IS cubic equation 238, 245 Einseinheiten 326 cubical norm 318 Eisenstein polynomial 345 cycle notation 33 Eisenstein's criterion 200 cyclic extension 235 elementary abelian group 26 - group 27 embedded component 385 - module 82 endomorphism 27, 80 cyclotomic polynomial 219ff. - ring 82, 135 endpoint 15 De Morgan's laws 70 enumerable 2 decomposition lemma 63, 361, 365 equipotent 1 Dedekind domain 115, 365ff., 390 equivalence xii, 67 Dedekind's lemma 206, 231 equivalent valuations 311 defining relation 26 - absolute values 315 degree of field extension 190 essential extension 129 -- polynomial 166 Euclid's Elements 193 Subject Index 461

Euclidean domain 351, 363, 371 - sum 248 - field 283 generating set 26 Euler function 161, 219, 248 generic point 379 - summation formula 248 going-up theorem 388 - criterion 248 Golod-Shafarevich theorem 177 Eulerian graph 23 graded algebra 165, 187 - partially ordered set 77 even Clifford algebra 261 graph 15 exact functor 111 - of a mapping 96 - sequence 84 Grassmann algebra 184 exceptional extension 414 greatest 8 exchange lemma 399 ground field 190 - property 397 group 25ff. expanded ideal 356 - action 29 exponent of a group 28, 441 - algebra 133 extension (field) 190 - word 26 exterior algebra 179, 263 external 40, 87 Hall's theorem 18 - 3 subgroup lemma 44 factor 36 Hamel basis 401 - theorem 34, 69, 85 Hasse invariant 296 faithful action 50 HCF highest common factor 347 - functor 67 height of p-radical element 410, 414 - representation 135 - of prime ideal 389 Fermat primes 242 Hensel's lemma 340f. Fermat's last theorem 362 henselization 341 fibre product, sum 88 hereditary ring 365, 372 field 80, 189 Hermitian conjugate 188 - of sets 70 Hilbert basis theorem 361 filtered algebra 187 - Nullstellensatz 392ff., 379 final object 68 - polynomial 175 finite 1, 335 - series 174ff. - character 12 - 'theorem 90' 438f. - field 223 homogeneous component 165f. finitely presented, related 106 homomorphism 26, 54, 80, 131, 167, 190 Fitting subgroup 48 Hopkins' theorem 145f. fixed field 211 hyperbolic pair, plane 267ff., 299 flat module 124, 358 - space 270 forgetful functor 66 formally real field 280 IBN invariant basis number 107, 110 fraction 335, 355 ideal 78, 80 fractional ideal 363f. - class group 370 Frattini subgroup 46f. - numbers 347, 362 free associative algebra 134, 168 idempotent 80 - field extensions 427 -law 53 - group 69 incidence algebra 157 - module 105f. - matrix 23 Frobenius mapping 203 inclusion-exclusion principle 160 full subcategory 65 independence property of tensor product fully invariant 94 120 function ring 377 index of a subgroup 28 functionally complete 73, 225 induced subgraph 15 functor 66ff. induction 13 fundamental involution 266 inductive ordered set 10 - theorem of algebra 202f., 285 inert subring 217 inf, infimum 51 Galois connexion 211 ff., 434 infinite set 2 - descent 437ff. initial object 68 - extension, group 211, 432 injective cogenerator 128 - field 223 - module 112 - theory, main theorem 212f., 434 inner automorphism 27 Gauss's lemma 217, 360, 425 - derivation 171 Gaussian extension 338 - product space 249 - integer 134, 152 inseparable degree 411 462 Basic Algebra

integers 79 Legendre polynomial 290 integral closure 332 - symbol 247 - domain 80 Leibniz's formula 173 - element 331 ff. length of a chain 61, 387 - extension 387 --- lattice 62 - ideal 364 Levitzki's theorem 146 interior multiplication 180 Lie algebra 43 intermediate value property 278 lie over 388 internal direct product 40 limit 275 -- sum 87 - ordinal 13 intersection graph 23 line complex, coordinates 185 invariant chain 36 linearly (in)dependent 105 inverse 25, 65 - disjoint 148,418 invertible element 80 local ring 335, 355, 396 - ideal 364 localization 355 (S-) inverting 335, 355 locally cyclic group 59, 436 involution 180,257 - finite group 28 irreducible algebraic set 379 -- partially ordered set 157 - element 63, 349 - nilpotent 382 - polynomial 193 loop 15 irredundant decomposition 383 lower bound 8 - intersection 143 - central series 40 isolated component 385 - segment 9 isometry 251 - semimodular lattice 77 isomorphic extensions 208 Liiroth's theorem 407 isomorphism 27, 29, 65, 80 - theorems 35ff., 80 Mac Lane's criterion 423 isotropic vector 257 marriage theorem 19 isotypic module 94 Maschke's theorem 162 matrix representation 134 Jacobi identity 43 - ring 97 Jacobson radical 48, 142f. matroid 445 join 52 maximal 8 join-(ir)reducible 63 maximum condition 60, 89 Jordan-Holder theorem 36 maxterm 72 meet 52 kernel 27, 80 meet-(ir)reducible 63, 383 Klein 4-group 26, 34, 328 minimal element 8 - quadric 185 - generating set 105 Konig's lemma 21 - polynomial 192 Konigsberg bridge problem 16 minimum condition 61, 89 Krasner's lemma 345 minterm 72 Kronecker's theorem 195 Mobius function 158, 248 Krull dimension 387 - inversion formula 158f. - intersection theorem 395 modular lattice 55 - topology 434 -law 28,55 Krull's theorem 90, 351 module 81ff. Krull-Schmidt theorem 96 monic polynomial 134 Kummer extension 441 monoid 30 Kurosh problem 178 - algebra 133, 168 Morita equivalence 100, 135f. Lagrange interpolation formula 104, 225, morphism 65 395 multiplication 79, 147 Lagrange resolvent 236 - table 133 Lagrange's theorem 28 multiplicative function 161 Laplace expansion 183 - representatives 326 lattice 52 - set 335, 351 Laurent polynomial 133, 166 multivector 179 law of quadratic reciprocity 248 LCM least common multiple 347 Nakayama's lemma 144f., 395 least element 8 natural duality 71 left, right exact 111 - homomorphism 34 -- inverse 109 - irrationality 233f. left-normed product 43 - transformation 66 Subject Index 463 negation 71 perfect closure 411 negative 273 - field 203 neutral element 25 - group 39 Newton-Fourier rule 324 permutation 26 nilideal 144 - group 32 nilpotence class 40 perspective intervals 56 nilpotent group, chain 39 PID principal ideal domain 80, 90, 372 - ideal 141 place 334 Plucker coordinates 185 nilradical 353 Poincare series 174 Noether normalization lemma 391 - 's theorem 32 Noether's equations 438 point 15 - problem 404 polynomial 133, 166 Noetherian induction 60 positive order set 273ff. - module 89 positive-definite 251, 285 - ring 89, 361 power of the continuum 7 non-defective form 302 power set 6 non-generator 46 preordering xi norm 153, 230 primary decomposition 383 - on Clifford algebra 266 - submodule 374, 382 normal basis theorem 439 prime avoidance lemma 384 - chain 36 - element 196, 310, 349 - closure 199 - ideal 196, 351 - equation 215 - subfield 189 - extension 198, 413 primitive element 223f., 228 - subgroup 27 - n-th root of 1 219 normalized valuation 309 - permutation group 50 normalizer 30 - polynomial 217 normed vector space 317 principal ideal 78, 80, 90 null sequence 275, 314 - valuation 308 Nullstellensatz 392ff. -- ring 311 principle of domination 322 object 65 -- inclusion-exclusion 160 one 79 product formula 191ff. opposite category 66 profinite group 434f. - ring 82 projective intervals 56 orbit 29 - module 112ff. order 332 projective-free ring 136 order of an element 26 proper orthogonal 256 - of a group 28 pullback action 86, 357 order set 273 - diagram 88 order-isomorphism 9, 54, 274 purely inseparable 156, 410 order-preserving mapping 54 - transcendental 404 order-type 9ff. pushout 88 ordered ring 272 Pythagorean field 298 ordinal (number) 11 orthogonal basis 254 quadratic extension 214 - group, transformation 256 - form, space 249 - idempotents 103 - residue 247 - sum, complement 252f. quadrature of circle 194 - vectors, space 251 quartic equation 238, 245 orthogonality relations 127 quasi compactness 380 Ostrowski's theorems 319f. quasi-Galois extension 413 outer derivation 171 quasi-inverse 143 quasiprimary ideal 386 p-adic integer, valuation 308, 311, 315, 369 quaternion algebra 148, 263f. p-dependent 446 - group 26, 50 p-group 30 quintic equation 238f., 245 p-radical extension 410 quiver 17 parallelogram law 35, 85 28 partially ordered monoid 63 path 17 R-module 81£. pentagon lattice 59 Rabinowitsch trick 393 464 Basic Algebra

radical of inner product space 251 - transcendental extension 405 -- ideal 353 simplicity of Alts 239f. -- ring 141ff., 353 singular form 251 - extension 235 skeleton 68 ramification index 337, 370 skew field 80 ramified 370 - symmetric matrix 298 Ramsey number, theorem 20 small category 65 rank of free algebra 173 socle 94 --- module 106 soluble (by radicals) 238 -- quadratic form 254 - group 39 real closed field 282 solution 376 reciprocal equation 247 source 65 recursive definition 13 spanning (set) 398 reduced ring 104, 421 - relation 400 reducible algebraic set 379 special orthogonal group 256 refinement 36, 61 Speiser's theorem 438 reflexion 257 regular field extension 425 Sperner's lemma 22 - mapping 396 spin group, representation 267 - part of quadratic space254 spinor kernel, norm 266f. - permutation group 34, 237 split exact 85 - representaton 134, 253 split inner product space 270 - quadratic form, space 251 - quadratic space 293 relation, relator 26 splitting field 197, 200 relative complement 56 - extension 430 relatively algebraically closed 405 stabilizer 29 represent 254 standard involution 180 residue class field 310, 356 Steinitz number 435 - degree 337 - criterion 228 resolvent 246 stochastic matrix, algebra 162 retraction 94 strongly regular ring 104 ring 79, 170, 347 structure constants 133 root 192 Sturm sequence, theorem 288ff. - tower 238 subring 80 rotation 256 sup, supremum 51 ruler-and-compass construction 194 superalgebra 168 supernatural number 435 saturated set 352 support 358 Schreier refinement theorem 37 Sylow subgroup theorems 37 Schroder-Bernstein theorem 4,77 Sylvester-Franke theorem 186 Schur's lemma 137ff. Sylvester's law of inertia 285 section 94 symbol homomorphism 305 self-regular 427 symbolic power 385 semi-Artinian module 129 symmetric difference 81 semidirect product 41 - functions 214 semigroup 30 - group 26 semilinear transformation 437 symmetry 257 semisimple module 91 symplectic basis, group 299f. - ring 137 - space 298 separable algebra 421 - closure 411 target 65 - degree 409 tensor algebra 169f. - element, polynomial 204f. - product 117ff. - extension 205, 432 theorem of the primitive element separably generated 423 228 separating transcendence basis 423 tiled ring 99 set 1 torsion (sub )module 90, 373 signature of a form 285 - element 90 similar algebras 152 - group 26 - quadratic forms 294 torsion-free 90, 376 simple extension 192, 228 totally ordered set 8 - group 34 - positive element 282 - module 91 trace 153, 230, 285 - ring 137 transcendence basis, degree 404 Subject Index 465 transcendental extension 192, 404 valency 17 transduction 187 valuation (ring) 308ff. transfinite induction 13, 61 Vandermonde determinant, matrix 232,238 - number 2 variety 379 transitive 29 versor 265 transitivity formulae 154, 230 vertex 15 transposition 33 transversal 28 weakly finite ring 107 tree 19 Wedderburn structure theorems 137ff. triangle inequality 273, 312 - nilpotence theorem 156 triangular matrix ring 99 - theorem on finite fields 226 trisection of an angle 194 well-ordered set 8 trivial absolute value 313 width of ordered set 17 - group 26 Witt (Grothendieck) ring 291ff. - ring 79 - group 271 - valuation 308 - identity 43 type component 94 - index, decomposition 270, 293 - invariant 296 UFD, unique factorization domain 217, 349, 359, - ring 270, 294 395 - cancellation theorem 269 UGN unbounded generating number llO - chain equivalence theorem 293 ultrametric inequality 313 - extension theorem 271 unary operator 70 uniformizer 310 Yoneda's lemma 78 unit 80 unit -element 25, 79 Zariski topology 380, 393 unital algebra 132 Zassenhaus lemma 37, 56 universal mapping property 67 Zech logarithm 224 - quadratic form 255 zero 192 upper bound 8 - element 25, 79 -- property 278 zerodivisor 80, 381 - central series 40 Zorn's lemma 10