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 of Algebraic 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 semi­primary 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 abelian group,

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 No Z Q

Q+ R C Urn Z(pOO) Z/(n) or Zin U(n) Fq Zp Qp = Zp[p-lj

Set Theory

o IXI Y'(X)

x\Y yX ~o

the natural numbers the natural numbers with 0 the integers the rational numbers the non-negative rational numbers the real numbers the complex numbers the group of m-th roots of unity the group of all pn_th roots of 1, for n = 1,2, ... the integers mod n 27 the group of units mod n 221 the field of q elements 224 the p-adic integers 315 the p-adic numbers 315

the empty set xi, 1 cardinal of the set X 2 power set (set of all subsets) of X 6 complement of Y in X xi set of all mappings from X to Y 5 aleph-null, the cardinal of N 2

453

454 Basic Algebra

Number Theory

max(a, b) min (a, b) alb (a, b) [a, b] 8ij

/-L(n) f/J(m) <l>m(x)

the larger of a, b the smaller of a, b a divides b highest common factor (HCF) of a and b least common multiple (LCM) of a and b Kronecker delta xi Mobius function 158 Euler function 161 cyclotomic polynomial 219

Group Theory

Symn Altn

sgn a Cn

Dm G' N<lG (G: H) GLn(R) SLn(R) Affn(k)

SP2m(k)

symmetric group of degree n 32 alternating group of degree n 33 sign of the permutation a 33 cyclic group of order n 27 dihedral group of order 2m 26

derived group of G 39 N is a normal subgroup of G 28 index of H in G 28 general linear group over a ring R special linear group over a ring R affine group over a field k symplectic group over a field k 299

Rings and Modules

mvn mv vn 9J1n(R) or Rn Lat(M) Hom(U, V) End(U)

U®V tM RO RX

Al Ann (X) Ass(M)

space of all m x n matrices over V 97 space of m-component columns over V (= mvl) 97 space of n-component rows over V (= I vn) 97

n x n matrix ring over R 97 lattice of all submodules of M 89 set of all homomorphisms from U to V 83 ring of all endomorphisms of U 83 tensor product of U and V 117 torsion submodule of M 90 opposite of the ring R 82 set of non-zero elements in R 80 augmented algebra of A 132 annihilator of X 84 assassinator of M 380

List of Notations

Supp(M) Rs or Rp Ja K[x] K[[xll K(X} sModR

$'(R)

TIM; UM; 'In(R)

Field Theory

[V: k] k(ex) k[ex] Gal(EjF) T(x) N(x) U1-V Ul.

(aJ, ... , an)

support of M 358 localization of R at 5 (or at the complement of p) 354f radical of an ideal a 353 polynomial ring on x over K 166 formal power series ring on x over K free K-algebra on X 134 category of (5, R)-bimodules 86 field of fractions of commutative integral domain R 428 direct product of modules 87 direct sum (coproduct) of modules 87 upper triangular matrices over R 133

dimension of the k-space V 190 field generated by ex over k 191 ring generated by ex over k 191 group of the Galois extension EjF 211 trace of x 153 norm of x 153 orthogonal sum of U and V 252f orthogonal complement of U 251 quadratic form (in diagonal form) 254

Categories (mappings resp. homomorphisms are understood)

Ens Gp Ab Rg Top Mod vec col Fun(I, d)

sets 65 groups 65 abelian groups 66 rings 65 topological spaces 65 modules 65 vector spaces 68 column vectors 68 functors from I to .91 69

455

Author Index

Abel, Niels Henrik (1802-29) 238 Abhyankar, Shreeram S. (1931-) 445 Adyan, Sergei 1. 28 Akgiil, M. II 0 Amitsur, Shimshon A. (1921-94) 146 Arf, Cahit (1910-) 303 Artin, Emil (1898-1962) 89, 139, 209, 216, 279,

283,285,316,319,432

Baer, Reinhold (1902-79) 116, 130 Banach, Stefan (1892-1945) 321 Bass, Hyman (1932-) 144 Becker, Eberhard 285 Bernstein, Felix (1878-1956) 4,77 Binet, Jacques P. M. (1786-1856) 182 Bolzano, Bernard (1781-1848) 2 Boole, George (181H4) 70ff. Bourbaki, Nicolas (1901-) 380 Brauer, Richard D. (1901-77) 152 Burali-Forti, Cesare (1861-1931) 6 Burnside, William (1852-1927) 28, 178

Cantor, Georg F.L.P. (1845-1918) 2,4, 7f., 14, 275

Capelli, Alfredo (1858-1916) 247 Cardano, Girolamo (1501-76) 238 Cartan, Henri P. (1904-) 258 Castelnuovo, Guido (1865-1952) 408 Cauchon, Gerard 446 Cauchy, Augustin Louis (1789-1857) 182, 275f. Cayley, Arthur (1821-95) 32, 135 Chevalley, Claude C (1909-84) 227, 333 Clifford, William K. (1845-79) 260,265 Cohen, Irving S. (1917-55) 395 Cohen, Paul J. (1934-) 7, 10 Cohn, Paul M. 108, 146, 185

De Morgan, Augustus (1806-71) 70 Dedekind, J.w.Richard (1831-1916) 2, 55, 115,

206,209,216,275,351, 362f., 365, 395 Deligne, Pierre Rene (1945-) 408 Descartes, Rene (1596-1650) 287 Dieudonne, Jean A. (1906-92) 137, 258 Dilworth, Robert P. (1914-93) 18 Diophantos (~250) 238 Dirichlet, Peter Gustav Lejeune

(1805-59) 2, 216, 222, 370

Eilenberg, Samuel (1913-98) 69 Eisenstein, Ferdinand Gotthold M. (1823-52)

199f. Erdos, Paul (1913-96) 21

457

Euler, Leonhard (1707-83) 17, 161,242, 247f. Faltings, Gerd (1954-) 362 Fermat Pierre de (1601-65) 242, 362, 371 Ferrari, Lodovico (1522-65) 238 Ferro, Scipio del (1465-1526) 238 Fitting, Hans (1906-38) 48 Flanders, Harley (1925-) 425 Franke, E. 186 Frattini, Giovanni (1852-1925) 44, 46f. Frobenius, F. Georg (1849-1917) 203,224

Galois, Evariste (1811-32) 206, 2I1ff., 216, 238f., 242, 244f.

Gauss, Carl F. (1777-1855) 134,217,242,248, 338

Gelfand, Izrael M. (1913-) 321 Godel, Kurt (1906-78) 7, 10 Golod, Evgenii S. (1935-) 176ff., 179 Goodearl, Kenneth R. (1945-) 129 Gottschalk, Walter H. 78 Grassmann, Hermann G. (1809-77) 184 Gregory, James (1638-75) 238 Grothendieck, Alexander (1928-) 291

Hahn, Hans (1879-1934) 321 Hall, Philip (1904-82) 18,44, 49 Halmos, Paul R.(1914-) 19 Hamel, Georg (1877-1954) 401 Hamilton, Sir William R.(l805-{)5) 148, 299 Harriot, Thomas (1560-1621) 287 Harrison, David K. (1931-90) 297 Hartley, Brian (1939-94) 146 Hasse, Helmut (1898-1980) 296 Hausdorff, Felix (1868-1942) 22,435 Hensel, Kurt (1861-1941) 311, 322, 340f. Hilbert, David (1863-1941) 174f., 221, 347, 361,

392 Hodges, Wilfrid A. (1941-) 60 Holder, Otto (1859-1937) 36 Hopkins, Charles (1902-39) 139, 145f. Huntingdon, Edward V. (1874-1952) 78

Ingleton, Aubrey W. (1921-2000) 445 Iversen, Birger 371

Jacobi, Carl Gustav Jacob (1804-51) 43 Jacobson, Nathan (1910-99) 142f. Jordan, Camille (1838-1922) 36

Kaplansky, Irving (1917-) 376 Klein, Avraham A. 146 Klein, C. Felix (1849-1925) 26, 185

458

Knebusch, Martin 285 Konig, Denes (1884-1944) 21 Konig, Gyula (Julius) (l849-1913) 5, 246 Krasner, Mark A. (l920-97) 345 Kronecker, Leopold (l823-9l) 195, 221, 246 Krull, Wolfgang (1899-1971) 90, 96, 395, 434 Kummer, Ernst-Eduard (1810-93) 347, 351,

362f., 441 Kuratowski, Kazimierz (1896-1980) 10 Kurosh, Aleksandr G. (1908-71) 178

Lagrange, Joseph L. (l736-1813) 28, 104,237 Laplace, Pierre S. Marquis de (l749-1827) 154,

184 Lasker, Emanuel (l868-194l) 380 Laurent, Pierre Alphonse (1803-54) 133, 166 Legendre, Adrien-Marie (l752-1833) 247f., 290 Leibniz, Gottfried Wilhelm, Freiherr von

(1646-1716) 173 Leicht, Johann B. 297 Lenstra Jr., Hendrik W. 404 Levitzki, Jacob (l904-56) 139, 146 Lindemann, C. L. Ferdinand von (1852-1939)

194 Liouville, Joseph (l809-82) 322 Lorenz, Falko 297 Lubell, David 22 Liiroth, Jakob (1844-1910) 407

Mac Lane, Saunders (1909-) 69, 423£, 445 Mahler, Kurt (1903-88) 328 Maschke, Heinrich (1853-1908) 162 Mazur, Stanislaw (l905-) 321 Merkuryev, A. 305 Moebius, August Ferdinand

(1790-1868) 158f. Moore, Eliakim Hastings (l862-1932) 224 Morita, Kiiti (l915-95) 100 Mumford, David B. (l937-) 311

Nagata, Masayoshi (l927-) 185, 395 Nakayama, Tadasi (1912-64) 144, 395 Newton, Sir Isaac (1642-1727) 325 Noether, A. Emmy (l882-1935) 61, 89, 139,265,

365£, 380, 391, 409, 438 Novikov, Petr S. 28

Ore, Oystein (1899-1968) 59 Ornstein, Donald S. 45 Ostrowski, Alexander (l893-1986) 319f.

Perlis, Sam (l913-) 142 Pfister, Albrecht (l934-) 305 Pierce, Richard S. (l927-92) 104 Pliicker, Julius (1801-68) 185 Poincare, J. Henri (l854-1912) 32, 174

Rabinowitsch, J. L. 393 Rados, G. 246 Ramsey, Frank Plumpton (1903-30) 19f. Riemann, Bernhard (l826-66) 203, 309 Roos, Jan-Erik. 96 Rota, Gian-Carlo (l932-99) 157

Basic Algebra

Ruffini, Paolo (1765-1822) 238 Russell, (Lord) Bertrand A. W. (1872-1970) 1,

10

Sarges, Heidrun 361 Scharlau, Winfried 283 Schmidt, Otto Yu. (l891-1956) 96 Schmidt, Friedrich Karl (l901-77) 425 Schreier, Otto (l901-29) 37, 56, 279 Schroder, F.W.K.Ernst (1841-1902) 4, 77 Schur, Issai (l875-1941) 137, 139,220 Seidenberg, Abraham(1916-) 389 Serre, Jean-Pierre (1926-) 174 Shafarevich, Igor R. (l923-) 176£. Sheffer, H. M. (l883-1964) 78 Sierpinski, Wad'aw (l882-1969) 22 Skolem, A. Thoralf (1887-1963) 265 Speiser, Andreas (l885-1970) 438 Sperner, Emanuel (l905-80) 22 Spitzlay, K.-E. 285 Steinitz, Ernst (l871-1928) 228, 238, 374, 432,

435 Stone, Marshall H. (l903-89) 76 Sturm, Jacques-Charles-Frans:ois (1803-55)

288ff. Swan, Richard G. (1933-) 404 Sylow, P. Ludvig M. (l832-1918) 37 Sylvester, James Joseph (l814-97) 186, 285f. Szekeres, George (l911-) 21

Tarski, Alfred (1902-83) 22 Tartaglia, Niccolo (1500-57) 238

Vahlen, Karl Theodor (1869-1945) 247 Vandermonde, Alexandre Theophile (l735-96)

232 Viete, Frans:ois (l540-1603) 238 Vinberg, Ernest B. 177 Voskresenskii, Valentin E. 404

Wall, Charles Terence Clegg (l936-) 296 Wantzel, Pierre L. (l814-48) 194 Warning, Ewald 227 Waterhouse, William C. 446 Weber, Heinrich (1842-1913) 221 Wedderburn Joseph H. Maclagan (l882-1948)

13 7ff., 156, 226 Weisner, Louis (l899-1988) 162 Whaples, George (l914-8l) 316 Whitney, Hassler (1907-89) 445 Wielandt, Helmut W. (1910-2001) 45,47 Wiles, Andrew (l953-) 362 Witt, Ernst (1911-91) 43, 226, 268ff., 291 ff.

Yoneda, Nobuo 78

Zafrullah, Muhammad (l942-) 394 Zariski, Oscar (l899-1986) 380, 393, 408 Zassenhaus, Hans J. (l912-91) 37,56 Zech, Theodor 224 Zelmanov, Efim I. (l955-) 28 Zermelo, Ernst F.F. (l871-1953) 10 Zorn, Max A. (1906-93) 10

Subject Index

Generally, non-X, un-X, in-X is listed under X.

abelian group 25 abelianization 67 absolute value 273, 312 absorptive law 53 acquaintanceship graph 16 acyclic 17 addition (mod 2) 81 additive function 173

- functor II 0 adjacency matrix 23 adjoint associativity 119, 123 affine group 227, 244 aleph 2 algebra 131 algebraic element 192

- equation 376 - extension 193, 403 - integer 134 - set 378

algebraically closed 201, 285, 431 - dependent 402

alternating form 256, 298, 301 - matrix 298 - group 33

anisotropic part 257, 270, 293 annihilator 84 anti-chain 8, 63f. anticommutative 166f. antiderivation 180 antihomomorphism 66, 83 approximation theorem 316, 369 archimedean absolute value 313

- ordering 277 Arf invariant 303 arrow 17 Artin's theorem 209 Artin-Schreier extension 444 Artin-Schreier theory 279ff. Artinian module, ring 89 assassinator 380 associated elements 349

- prime ideal 380 associative law 25, 53 atom 75, 193, 349 atomic Boolean algebra 78

- domain 350

459

augmentation ideal 132, 167, 293 augmented algebra 132 automorphism 27, 80, 211 axiom of choice 10,60

Baer's criterion 116 balanced mapping 122 basis 105, 398

- theorem for abelian groups 38 bidual 126 bifunctor 87 bilinear form 249ff.

- mapping 117, 122 bimodule 83 binary form 249 Binet-Cauchy identity 182 Boolean algebra 70, 134

- polynomial 71 - ring 80

Brauer group 152, 296 Burnside problem 28, 178

cancellation 31 cardinal (number) Iff. Cartan-Dieudonne theorem 258 Castelnuovo-Zariski theorem 408 casus irreducibilis 247 category 65 Cauchy sequence 275, 314 Cayley's theorem 32, 135, 214 central chain 39

- simple algebra 150 centralizer 30, 149 centrally primitive idempotent 103 centre 30, 131 chain 8

- condition 60 character (group) 125 characteristic of field 189

- - prime ideal 297 - - ring 80, 104 - function 6 - polynomial 153, 175 - subgroup 46

Chevalley's lemma 333 chief factor, series 36

460

Chinese remainder theorem 102, 115 class 65

- equation 30 - of nilpotence 40

classical logic 71 Clifford algebra 260

- group 265f. closed set 378ff. cofinal 14 cofinite subset 70 1. S. Cohen's theorem 395 coimage, cokernel 85 comaximal ideals 102 comma category 69 commutative diagram 85

-law 25,53 - ring 79

commutator 39, 42 companion matrix 155 complement 56, 92 complementary graph 16 complete lattice 55

- ordered field 275 - space 314

completely primary ring 104, 187, 386 completion 277, 315 composite of fields 428 composition series 36 compound matrix 182 concrete category 66 conductor 391 cone 279f. congruent matrices 250 conjugacy class 30 conjugate 30, 42, 199,213 conjunctive normal form 72 connected graph 19 conorm mapping 370 consistent system 377 constant 171,415 continuum hypothesis 7 contracted ideal 356 converge 275 co(ntra)variant functor 66 coordinate ring 377 core of a field 280 coset (space) 27ff. countable 2 cubic equation 238, 245 cubical norm 318 cycle notation 33 cyclic extension 235

- group 27 - module 82

cyclotomic polynomial 219ff.

De Morgan's laws 70 decomposition lemma 63, 361, 365 Dedekind domain 115, 365ff., 390 Dedekind's lemma 206, 231 defining relation 26 degree of field extension 190

- - polynomial 166

- - rational function 405 - - symmetric group 26

Delian problem 194 denominator 335, 354f.

dense embedding 275, 393 - functor 67

density principle 228 dependence relation 397ff. derivation 171, 415ff. derivative 204 derived group, series 39 determinant 182

- of a form 252 diagonal argument 8 diamond lattice 57 dicyclic group 50 different 395 digraph 17 dihedral group 26, 50 Dilworth's theorem 18 dimension 140, 249, 387, 403

- index 295 direct power 87

- product 27, 40, 87 - - of rings 101 - sum of modules 87

directed graph 52 - system 64, 202

Dirichlet box principle 2 Dirichlet's theorem 222 discrete absolute value 313

- rank 1 valuation 308 discriminant 231, 263 disjunctive normal form 72 distributive lattice 59, 96

- law 59, 79, 119 divisible module 116 division algebra 133, 150

- ring 80 dominate 332 dual basis lemma 114

- categories 67 - group 125 - homomorphism 71

duality 67f.

edge IS Einseinheiten 326 Eisenstein polynomial 345 Eisenstein's criterion 200 elementary abelian group 26 embedded component 385 endomorphism 27, 80

- ring 82, 135 endpoint 15 enumerable 2 equipotent 1 equivalence xii, 67 equivalent valuations 311

- absolute values 315 essential extension 129 Euclid's Elements 193

Basic Algebra

Subject Index

Euclidean domain 351, 363, 371 - field 283

Euler function 161, 219, 248 - summation formula 248 - criterion 248

Eulerian graph 23

even Clifford algebra 261 exact functor 111

- sequence 84 exceptional extension 414 exchange lemma 399

- property 397 expanded ideal 356 exponent of a group 28, 441 extension (field) 190 exterior algebra 179, 263 external 40, 87

factor 36 - theorem 34, 69, 85

faithful action 50 - functor 67 - representation 135

Fermat primes 242 Fermat's last theorem 362 fibre product, sum 88 field 80, 189

- of sets 70 filtered algebra 187 final object 68 finite 1, 335

- character 12 - field 223

finitely presented, related 106 Fitting subgroup 48 fixed field 211 flat module 124, 358 forgetful functor 66 formally real field 280 fraction 335, 355 fractional ideal 363f. Frattini subgroup 46f. free associative algebra 134, 168

- field extensions 427 - group 69 - module 105f.

Frobenius mapping 203 full subcategory 65 fully invariant 94 function ring 377 functionally complete 73, 225 functor 66ff. fundamental involution 266

- theorem of algebra 202f., 285

Galois connexion 211 ff., 434 - descent 437ff. - extension, group 211, 432 - field 223 - theory, main theorem 212f., 434

Gauss's lemma 217, 360, 425 Gaussian extension 338

- integer 134, 152

- sum 248 generating set 26 generic point 379 going-up theorem 388 Golod-Shafarevich theorem 177 graded algebra 165, 187

- partially ordered set 77 graph 15

- of a mapping 96 Grassmann algebra 184 greatest 8 ground field 190 group 25ff.

- action 29 - algebra 133 - word 26

Hall's theorem 18 - 3 subgroup lemma 44

Hamel basis 401 Hasse invariant 296 HCF highest common factor 347 height of p-radical element 410, 414

- of prime ideal 389 Hensel's lemma 340f. henselization 341 hereditary ring 365, 372 Hermitian conjugate 188 Hilbert basis theorem 361

- Nullstellensatz 392ff., 379 - polynomial 175 - series 174ff. - 'theorem 90' 438f.

homogeneous component 165f. homomorphism 26, 54, 80, 131, 167, 190 Hopkins' theorem 145f. hyperbolic pair, plane 267ff., 299

- space 270

IBN invariant basis number 107, 110 ideal 78, 80

- class group 370 - numbers 347, 362

idempotent 80 -law 53

incidence algebra 157 - matrix 23

inclusion-exclusion principle 160 independence property of tensor product

120 index of a subgroup 28 induced subgraph 15 induction 13 inductive ordered set 10 inert subring 217 inf, infimum 51 infinite set 2 initial object 68 injective cogenerator 128

- module 112 inner automorphism 27

- derivation 171 - product space 249

inseparable degree 411

461

462

integers 79 integral closure 332

- domain 80 - element 331 ff. - extension 387 - ideal 364

interior multiplication 180 intermediate value property 278 internal direct product 40

- - sum 87 intersection graph 23 invariant chain 36 inverse 25, 65 invertible element 80

- ideal 364 (S-) inverting 335, 355 involution 180,257 irreducible algebraic set 379

- element 63, 349 - polynomial 193

irredundant decomposition 383 - intersection 143

isolated component 385 isometry 251 isomorphic extensions 208 isomorphism 27, 29, 65, 80

- theorems 35ff., 80 isotropic vector 257 isotypic module 94

Jacobi identity 43 Jacobson radical 48, 142f. join 52 join-(ir)reducible 63 Jordan-Holder theorem 36

kernel 27, 80 Klein 4-group 26, 34, 328

- quadric 185 Konig's lemma 21 Konigsberg bridge problem 16 Krasner's lemma 345 Kronecker's theorem 195 Krull dimension 387

- intersection theorem 395 - topology 434

Krull's theorem 90, 351 Krull-Schmidt theorem 96 Kummer extension 441 Kurosh problem 178

Lagrange interpolation formula 104, 225, 395

Lagrange resolvent 236 Lagrange's theorem 28 Laplace expansion 183 lattice 52 Laurent polynomial 133, 166 law of quadratic reciprocity 248 LCM least common multiple 347 least element 8 left, right exact 111

- - inverse 109 left-normed product 43

Legendre polynomial 290 - symbol 247

Leibniz's formula 173 length of a chain 61, 387

- - - lattice 62 Levitzki's theorem 146 Lie algebra 43 lie over 388 limit 275

- ordinal 13 line complex, coordinates 185 linearly (in)dependent 105

- disjoint 148,418 local ring 335, 355, 396 localization 355 locally cyclic group 59, 436

- finite group 28 - - partially ordered set 157 - nilpotent 382

loop 15 lower bound 8

- central series 40 - segment 9 - semimodular lattice 77

Liiroth's theorem 407

Mac Lane's criterion 423 marriage theorem 19 Maschke's theorem 162 matrix representation 134

- ring 97 matroid 445 maximal 8 maximum condition 60, 89 maxterm 72 meet 52 meet-(ir)reducible 63, 383 minimal element 8

- generating set 105 - polynomial 192

minimum condition 61, 89 minterm 72 Mobius function 158, 248

- inversion formula 158f. modular lattice 55

-law 28,55 module 81ff. monic polynomial 134 monoid 30

- algebra 133, 168 Morita equivalence 100, 135f. morphism 65 multiplication 79, 147

- table 133 multiplicative function 161

- representatives 326 - set 335, 351

multivector 179

Nakayama's lemma 144f., 395 natural duality 71

- homomorphism 34 - irrationality 233f. - transformation 66

Basic Algebra

Subject Index

negation 71 negative 273 neutral element 25 Newton-Fourier rule 324 nilideal 144 nilpotence class 40 nilpotent group, chain 39

- ideal 141

nilradical 353 Noether normalization lemma 391 Noether's equations 438

- problem 404 Noetherian induction 60

- module 89 - ring 89, 361

non-defective form 302 non-generator 46 norm 153, 230

- on Clifford algebra 266 normal basis theorem 439

- chain 36 - closure 199 - equation 215 - extension 198, 413 - subgroup 27

normalized valuation 309 normalizer 30 normed vector space 317 null sequence 275, 314 Nullstellensatz 392ff.

object 65 one 79 opposite category 66

- ring 82 orbit 29 order 332 order of an element 26

- of a group 28 order set 273 order-isomorphism 9, 54, 274 order-preserving mapping 54 order-type 9ff. ordered ring 272 ordinal (number) 11 orthogonal basis 254

- group, transformation 256 - idempotents 103 - sum, complement 252f. - vectors, space 251

orthogonality relations 127 Ostrowski's theorems 319f. outer derivation 171

p-adic integer, valuation 308, 311, 315, 369 p-dependent 446 p-group 30 p-radical extension 410 parallelogram law 35, 85 partially ordered monoid 63 path 17 pentagon lattice 59

perfect closure 411 - field 203 - group 39

permutation 26 - group 32

perspective intervals 56 PID principal ideal domain 80, 90, 372 place 334 Plucker coordinates 185 Poincare series 174

- 's theorem 32 point 15 polynomial 133, 166 positive order set 273ff. positive-definite 251, 285 power of the continuum 7 power set 6 preordering xi primary decomposition 383

- submodule 374, 382 prime avoidance lemma 384

- element 196, 310, 349 - ideal 196, 351 - subfield 189

primitive element 223f., 228 - n-th root of 1 219 - permutation group 50 - polynomial 217

principal ideal 78, 80, 90 - valuation 308 - - ring 311

principle of domination 322 - - inclusion-exclusion 160

product formula 191ff. profinite group 434f. projective intervals 56

- module 112ff. projective-free ring 136 proper orthogonal 256 pullback action 86, 357

- diagram 88 purely inseparable 156, 410

- transcendental 404 pushout 88 Pythagorean field 298

quadratic extension 214 - form, space 249 - residue 247

quadrature of circle 194 quartic equation 238, 245 quasi compactness 380 quasi-Galois extension 413 quasi-inverse 143 quasiprimary ideal 386 quaternion algebra 148, 263f.

- group 26, 50 quintic equation 238f., 245 quiver 17 quotient group 28

R-module 81£. Rabinowitsch trick 393

463

464

radical of inner product space 251 - - ideal 353 - - ring 141ff., 353 - extension 235

ramification index 337, 370 ramified 370 Ramsey number, theorem 20 rank of free algebra 173

- - - module 106 - - quadratic form 254

real closed field 282 reciprocal equation 247 recursive definition 13 reduced ring 104, 421 reducible algebraic set 379 refinement 36, 61 reflexion 257 regular field extension 425

- mapping 396 - part of quadratic space254 - permutation group 34, 237 - representaton 134, 253 - quadratic form, space 251

relation, relator 26 relative complement 56 relatively algebraically closed 405 represent 254 residue class field 310, 356

- degree 337 resolvent 246 retraction 94 ring 79, 170, 347 root 192

- tower 238 rotation 256 ruler-and-compass construction 194

saturated set 352 Schreier refinement theorem 37 Schroder-Bernstein theorem 4,77 Schur's lemma 137ff. section 94 self-regular 427 semi-Artinian module 129 semidirect product 41 semigroup 30 semilinear transformation 437 semisimple module 91

- ring 137 separable algebra 421

- closure 411 - degree 409 - element, polynomial 204f. - extension 205, 432

separably generated 423 separating transcendence basis 423 set 1 signature of a form 285 similar algebras 152

- quadratic forms 294 simple extension 192, 228

- group 34 - module 91 - ring 137

- transcendental extension 405 simplicity of Alts 239f. singular form 251 skeleton 68 skew field 80

- symmetric matrix 298 small category 65 socle 94 soluble (by radicals) 238

- group 39 solution 376 source 65 spanning (set) 398

- relation 400 special orthogonal group 256 Speiser's theorem 438

Sperner's lemma 22 spin group, representation 267 spinor kernel, norm 266f. split exact 85 split inner product space 270

- quadratic space 293 splitting field 197, 200

- extension 430 stabilizer 29 standard involution 180 Steinitz number 435

- criterion 228 stochastic matrix, algebra 162 strongly regular ring 104 structure constants 133 Sturm sequence, theorem 288ff. subring 80 sup, supremum 51 superalgebra 168 supernatural number 435 support 358 Sylow subgroup theorems 37 Sylvester-Franke theorem 186 Sylvester's law of inertia 285 symbol homomorphism 305 symbolic power 385 symmetric difference 81

- functions 214 - group 26

symmetry 257 symplectic basis, group 299f.

- space 298

target 65 tensor algebra 169f.

- product 117ff. theorem of the primitive element

228 tiled ring 99 torsion (sub )module 90, 373

- element 90 - group 26

torsion-free 90, 376 totally ordered set 8

- positive element 282 trace 153, 230, 285 transcendence basis, degree 404

Basic Algebra

Subject Index

transcendental extension 192, 404 transduction 187 transfinite induction 13, 61

- number 2 transitive 29 transitivity formulae 154, 230 transposition 33 transversal 28 tree 19 triangle inequality 273, 312 triangular matrix ring 99 trisection of an angle 194 trivial absolute value 313

- group 26 - ring 79 - valuation 308

type component 94

UFD, unique factorization domain 217, 349, 359, 395

UGN unbounded generating number llO ultrametric inequality 313 unary operator 70 uniformizer 310 unit 80 unit -element 25, 79 unital algebra 132 universal mapping property 67

- quadratic form 255 upper bound 8

- - property 278 - central series 40

valency 17 valuation (ring) 308ff. Vandermonde determinant, matrix 232,238 variety 379 versor 265 vertex 15

weakly finite ring 107 Wedderburn structure theorems 137ff.

- nilpotence theorem 156 - theorem on finite fields 226

well-ordered set 8 width of ordered set 17 Witt (Grothendieck) ring 291ff.

- group 271 - identity 43 - index, decomposition 270, 293 - invariant 296 - ring 270, 294 - cancellation theorem 269 - chain equivalence theorem 293 - extension theorem 271

Yoneda's lemma 78

Zariski topology 380, 393 Zassenhaus lemma 37, 56 Zech logarithm 224 zero 192

- element 25, 79 zerodivisor 80, 381 Zorn's lemma 10

465