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Bibliography 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 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 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 <l>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 cyclic group of order n 27 Dm dihedral group of order 2m 26 G' derived group of G 39 N<lG N is a normal subgroup of G 28 (G: H) index of H in G 28 GLn(R) general linear group over a ring R SLn(R) special linear group over a ring R Affn(k) affine group over a field k SP2m(k) symplectic group over a field k 299 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.
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