Algebra I*

obligatory courses

(in Polish) Oczekuje się dobrej znajomości zagadnień ujętych w sylabusach przedmiotów Geometria z algebrą liniową I oraz Wstęp do matematyki.

This is an extended version of the course Algebra 1; enriched by additional material on group and ring theory.

Fundamental algebraic structures: groups, commutative rings with 1 and fields. Group theory: normal subgroups, factor groups, group actions on sets, information about Sylow’s theorems and the classification of finitely generated abelian groups. Ring theory: divisibility, unique factorization, the notions of an ideal and of the factor ring. Field theory: field extensions obtained by adding roots of a polynomial and information on the existence of the algebraic closure.

1. Basic definitions and examples: groups, rings, fields and their substructures. Homomorphisms. Examples: groups of transformations (including permutation groups, dihedral groups, linear groups GL(n;K), O(n), SO(n)). Rings: the ring of integers, polynomial rings of one and of several variables, rings of powers series. Fields: Q, R, C, Zp.

2. Foundations of group theory: order of an element, generating sets, cyclic groups and their properties. Direct product and its inner characterization. Classification of finitely generated abelian groups (without proofs). Cosets, Lagrange’s theorem and applications.

3. Group actions on sets. Orbits, isotropy groups, equivariant isomorphism between an orbit and the corresponding set of cosets. Applications: Cayley’s theorem, Cauchy’s theorem, inner automorphisms and conjugacy classes, decomposition of a permutation into disjoint cycles, the centre of a finite p-group. Conjugacy classes in permutation groups.

4. The kernel of a homomorphism, normal subgroup, factor group. Isomorphism theorem. Derived subgroup and abelianization. Simple groups, alternating groups.

5. Sylow’s theorems (formulation and simple examples only).

6. Commutative rings. Special elements (units, zero divisors, nilpotent elements). Field of quotients of a commutative domain. The kernel of a homomorphism, ideals, factor rings, isomorphism theorem. Prime ideals and maximal ideals. Examples.

7. Ideals in the ring of integers and in the polynomial ring with coefficients in a field. Bezout’s theorem. Polynomial functions. Principal ideal domains and examples. Euclidean domains.

8. Irreducible elements and prime elements. Unique factorization domains. Examples. Unique factorization in principal ideal domains. Unique factorization in polynomial rings – Gauss lemma (formulation only). Eisenstein criterion.

9. Field extensions obtained by adding a root of a polynomial. Prime subfields. Examples of constructions of finite fields. Information on the existence of the algebraic closure.

1. M. Kargapolov, J. Merzljakov, Foundations of Group Theory, Springer-Verlag, 1979

2. L. Rowen, Algebra, Rings and Fields, A. K. Peters, Ltd., 1994

3. B.L.Van der Waerden, Algebra, Springer-Verlag, 1991

1. Knows the notions of groups, rings and fields and homomorphisms of these structures. Is able to give and specify different examples of such structures.

2. Knows fundamental constructions of groups, Lagrange's theorem and its proof. Is able to describe elements in the group generated by a set, to prove the structure theorem of cyclic groups and knows the formulation of the theorem describing finitely generated abelian groups.

3. Knows the notions of an action of a group on a set, notions of orbits and stabilizers and relations between them. Knows applications, in particular: Cayley's theorem, Cauchy's theorem and the theorem asserting that the center of a non-trivial finite p-group is non-trivial.

4. Knows the notions of normal subgroup and the factor group. Is able to describe normal subroups in distinguished examples of groups. Knows and is able to apply the isomorphism theorem. Knows the notions of the derived subgroup and the abelianization of a group.

5. Knows the notions of elements of special types rings (zero divisors, nilpotents, invertible elements) and is able to describe them in specified examples of rings. Knows the notion of an ideal. Is able to describe elements of the ideal generated by a set. Knows the notions of prime and maximal ideals as well as the relationship between them and their characterizations in terms of factor rings.

6. Knows the notions of principal ideal rings. Is able to describe ideals in the ring of integers and in polynomial rings in one indeterminate over fields. Knows the notion of Euclidean domains and is able to prove that they are principal ideal domains. Knows examples of Euclidean domains, including the ring of Gaussian integers.

7. Knows the notions of irreducible and prime elements, relationships between them and the definition of unique factorization domains. Is able to provide examples of important classes of unique factorization domains as well as examples of domains that are not unique factorization domains.

8. Knows the notion of an algebraic element and is able to construct a field extension of a field by a root of a given polynomial with coefficients in that field. Knows the notion of the algebraic closure. Is able to describe possible cardinalities of finite fields and to justify this description.

9. Knows number theoretic applications of the notions and results presented at the lectures.