Università Roma Tre

Sets and maps. Relations of equivalence. Natural numbers . Peano's Axioms. Principle of Induction. Well Ordering Axiom. Construction (from the set of natural numbers N) of the set of integers Z and of the set of rational numbers Q. Basic properties of complex numbers, C. Theory of divisibility in Z, Euclidean algorithm, GCD. Definition and examples of the main algebraic structures: groups, rings and fields. Group of units of a ring. Permutation groups. The ring of number classes modulo n. Linear congruences. Polynomial rings with numerical coefficients: definitions, basic properties, theory of divisibility, irreducibility criteria, Gauss Lemma.

Suggested references
-- G.M. Piacentini Cattaneo, Algebra, un approccio algoritmico, Decibel-Zanichelli, (1996)
-- M. Fontana - S. Gabelli, Insiemi, numeri e polinomi. Primo ciclo di lezioni del corso di Algebra con esercizi svolti. CISU, (1989)
-- M. Fontana - S. Gabelli, Esercizi di Algebra, Aracne, (1993)

Further references (useful also for AL210)
-- R.B.J.T. Allenby, Rings, Fields and Groups, Edward Arnold, (1991)
-- M. Artin, Algebra, Prentice-Hall, (1991)
-- L. Childs, Algebra, un'introduzione concreta, Springer (1983) e ETS (1989)
-- D. Dikranjan - M.S. Lucido, Aritmetica e algebra, Liguori Editore, (2007)
-- J. B. Fraleigh, A First Course in Abstract Algebra, 7th ed. Reading, MA, Addison-Wesley, (2002)
-- S. Gabelli - F. Girolami, Anelli di Polinomi, Dispense .pdf
-- J.A. Gallian, Contemporary Abstract Algebra (5th edition), Houghton-Mifflin (1998)
-- I.N. Herstein, Algebra, Editori Riuniti, (2003)
-- A.C. Hibbard –K.M. Levasseur, Exploring Abstract Algebra with Mathematica, TELOS/Springer-Verlag (1999)
-- T.W. Hungerford, Algebra, Graduate Texts in Mathematics, Springer (1980)
-- I. Kleiner, A History of Abstract Algebra, Birkhäuser (2007)
-- S. Lang, Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), Springer-Verlag, (2002)
-- B. Scimemi, Algebretta, Decibel e Zanichelli, (1989)
-- B. L. van der Waerden, Algebra (new ed. of Moderne Algebra) voll. I and II, Springer, (1959)
-- B.L. van der Waerden, A History of Algebra, Springer (1990)
-- V.S. Varadarajan, Algebra in Ancient and Modern Times, American Mathematical Society (1998)