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https://www.uniud.it/it/didattica/info-didattiche/regolamento-didattico-del-corso/L-matematica/all-B2
a written test lasting about three hours, which consists in the resolution of some exercises;
an oral test, accessible after passing the written test, which consists of questions regarding the theory.
The final mark is computed from the mark of the written test and the mark of the oral test; in particular, to pass the exam, it is necessary that the marks of both parts are sufficient.
The exam can be split in two parts corresponding to the program of the first and of the second semester respectively.
The course can be taught in English on the proposal of the competent teaching structure.
The course concerns arguments of abstract algebra and develops the abstract language of algebra, by introducing basic notions of group theory, ring theory and field theory. Special attention is payed to the study of symmetric groups, linear groups and the ring of polynomials.
Sector-specific skills
Knowledge and understanding:
Understand and know the fundamental concepts of group theory.
Understand and know the fundamental concepts of ring theory and field theory.
Use correctly the algebraic language.
Applying knowledge and understanding:
Apply the theory to solve the given exercises.
Identify and formalize the algebraic structures.
Cross-sectoral skills/soft skills
Making judgments:
Identify the algebraic techniques most suitable for solving the assigned problems.
Judge independently the correctness of the proof of a theorem.
Communication Skills:
Introduce clearly and logically the learned topics.
Communicate properly the proof of a theorem or the resolution of an exercise.
Learning skills:
Acquire an appropriate method of study to learn the teaching matters and new related topics.
Study independently starting with the recommended bibliography.
Algebraic structures: semigroups, monoids, groups.
First notions in group theory: groups and subgroups, order of an element.
Cosets of a subgroup, Lagrange Theorem.
Normal subgroups, quotient group.
Homomorphisms, Homomorphism Theorems, Correspondence Theorem.
Direct products of groups.
Cyclic groups, Structure Theorem of finite abelian groups.
Permutation groups, Cayley Theorem.
Quaternions group and Dihedral group.
Automorphisms groups.
Classes Equation.
Sylow Theorems.
Second semester.
Basic notions in ring theory: subrings, ideals, quotient ring, homomorphisms, direct products, matrices.
The Quaternions.
Maximal ideals and prime ideals of a commutative ring, Krull Theorem.
Integral domains, quotients field.
Polynomial ring on a field: Ruffini Theorem, division algorithm.
Irreducible and prime elements, unique factorization domains, principal ideal domains, euclidean domains.
Gauss integers.
First notions in field theory, finite fields, roots of unity.
Irreducible polynomials, factorization, Eisenstein criterion.
Field extensions, algebraic and transcendental elements.
Algebraically closed fields.
