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. At the end of the course the student will be able to use an appropriate algebraic language and a proper mathematical formalism to report on the arguments presented.
Course contents summary
The course is an introduction to different aspects of Algebra. It starts with the set theory : notations, operations between sets and their main properties, correspondences and functions between sets, relations in a set, quotient set and construction of Z and Q. It continues with the elementary arithmetic on Z: division algorithm., G.C.D. and Bézout’s identity, prime numbers and fundamental theorem of Arithmetic, the ring of residue classes modulo n, Zp fields, linear congruences and solving them, the Chinese remainder theorem, Euler’s function and Euler’s theorem. The second part of the course is aimed to the discussion of algebraic structures with one or two operations. In particular, the elementary topics of group theory and the first examples are introduced: Sn symmetric group and dihedral groups, cosets modulo a subgroup and Lagrange’s theorem, isomorphisms between groups and Cayley’s theorem, homomorphisms, normal subgroups, quotient group and homomorphism theorem, cyclic groups and group action on a set.
In the third part of the course the arithmetic of Z is generalized to polynomials and to other domains: construction and properties of the polynomial ring in one variable with coefficients in a field, questions of irreducibility in polynomials in C, R , Q , Zp, euclidean domains, principal domains and factorisation domains. The course ends with a discussion of the fields as abstract structures which include the cases C, R, Q and Zp: algebraic extensions of finite degree, field extension theorem, splitting field of a polynomial and finally some elementary properties and the construction of finite fields.
• 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.
• 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.
• 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.
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 - Undergraduete Texts in Mathematics, Springer
G.M. Piacentini Cattaneo, ALGEBRA, Un approccio algoritmico - Zanichelli Editore
The preferred teaching tool for the development of such knowledge are the lectures and the exercises. The note taking is seen as part of the learning process. The sessions of exercises are seen as a very effective and essential tool in Algebra, where understanding is acquired through practice and not through mere storage. Exercises are very often proposed to be done by the students themselves: through the execution the students can be encouraged to explore the limits of their capabilities.
Assessment methods and criteria
The assessment of learning is done in classic way, through the evaluation of a written test and an oral interview. The student can also perform 4 written exams during the course, that are valid for the purposes of passing the written test.
In written exams, through the exercises, the student must demonstrate basic knowledge related to the study of algebraic structures such as groups, rings and fields, with particular consideration to the study of the polynomial rings and the properties of finite fields. In addition, the student will be required to deal, in an autonomous way, with problems related to the theories studied.
In the colloquium, the student must be able to independently conduct demonstrations related to the intrinsic properties of the studied structures using an appropriate algebraic language and a proper mathematical formalism.