INTRODUCTION TO ALGEBRA
Learning outcomes of the course unit
Know the language of the set theory to correctly formulate mathematical statements and precisely construct simple demonstrations. Know how to work with equivalence classes and quotient sets. Know how to abstractly recognise the main algebraic structures and their properties, especially groups, rings, integral domains and fields. Know how to concretely work in the ring of integers, ring of residue classes and rings of polynomials with coefficients in C,R,Q and field of residue classes modulo a prime.
Course contents summary
• Set theory: notations, characteristic representation, set families. Operations between sets and their main properties. Correspondences between sets. Functions between sets: injective, surjective and bijective functions. Composition of functions and relative properties.
• Relations in a set: relations of order and equivalence. Quotient set.
• Congruences: properties and applications. Solving linear congruences and the Chinese remainder theorem. Euler’s function and Euler’s theorem. Prime numbers.
• Algebraic structures: definition of an internal operation on a set. Neutral element and symmetrizable elements. Properties of structures with one or two operations.
• Groups: definition and first examples. Sn symmetric group. Dihedral groups. Coset modulo some subgroup and Lagrange’s Theorem Isomorphisms between groups and Cayley’s Theorem. Homomorphisms, conjugacy. Normal subgroups. Quotient group. Homomorphism theorem. Cyclic groups. Group action on a set: orbits and stabilisers. Permutation groups. Burnside’s formula. Cauchy’s theorem and Sylow’s theorems. Overview of the classification of finite abelian groups.
• Z ring of integers: properties of Z. Division algorithm. G.C.D. and Bézout’s identity. Prime numbers and properties. Fundamental theorem of arithmetic. The ring of residue classes modulo n. Invertibility of residue classes. Zp fields. Applications of congruences. Fermat’s little theorem and applications. Linear congruences and solving them. The Chinese remainder theorem. Euler’s function and Euler’s theorem.
• The polynomial ring: definition and construction of the polynomial ring in a variable with coefficients in a ring or a field. Properties of the polynomial ring in a variable with coefficients in a field: division between polynomials, G.C.D., factorisation. Questions of irreducibility in polynomials in C, R , Q , Zp. Overview of cyclotomic and symmetric polynomials and polynomials in more than one variable.
• Rings: ring as an abstract structure which includes the cases Z, Zn and polynomial ring. Definitions and examples, general properties. Subrings. Homomorphisms between rings. Ideals. Quotient ring. Homomorphism theorems and isomorphism between rings. Ideal generated by a subset. Prime and maximal ideals and relative theorems. Ideals of Z and the polynomial ring in a field. Congruence module a polynomial. Quotient field of an integral domain. Euclidean domains, principal domains and factorisation domains. The characteristic of an integral domain.
• Fields: definitions, examples and general properties. Field as an abstract structure which includes the cases C, R, Q and Zp. Algebraic extensions of finite degree. Algebraic and transcendental elements. Minimum polynomial of an algebraic element. Field extension theorem. Splitting field of a polynomial. Elementary properties and construction of finite fields. Overview of Galois extensions and solvability of a polynomial equation by radicals, Abel-Ruffini theorem.
S.Franciosi, F.de Giovanni, ELEMENTI DI ALGEBRA - Aracne Editrice
M.Curzio, P.Longobardi,M.May, LEZIONI DI ALGEBRA - Liguori Editore
J. Stillwell, ELEMENTS OF ALGEBRA - Undergraduate Texts in Mathematics, Springer
During the course 4 written tests will be given valid for passing the written examination.