Algebra I

obligatory courses

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

The aim of the lecture is to introduce fundamental algebraic structures: groups, commutative rings with 1 and fields, and to discuss their basic properties. The properties of rings are presented as a natural extension of the properties of the ring of integers and the ring of polynomials over a field. In particular,

the following topics are discussed: divisibility, unique factorization, the notions of an ideal and of the quotient ring. The part of the lecture devoted to fields includes field extensions obtained by adding roots of a polynomial and the

information on the algebraic closure. The construction of the quotient field of a domain is presented. The part of the lecture concerning group theory covers basic properties of groups but it also includes information about the classification of finitely generated abelian groups and about actions of finite groups on sets and their simplest applications.

1. The ring of integers Z and the ring of integers modulo m, Z_m. The definition of a commutative ring with 1. Subring. Units, zero divisors, domain. Divisibility, irreducible elements. The Euclidean algorithm in Z, greatest common divisor. [1 lecture]

2. Ring of polynomials in one variable over a field, ring of polynomials in several variables. Divisibility and the Euclidean algorithm in k[x]. The definition of a unique factorization domain (UFD). Theorem: Z and k[x] are UFD. Polynomials in one variable: polynomial functions, roots of polynomials, Remainder and Factor theorems. Irreduciibility of polynomials, the Eisenstein criterion and reduction of coefficients. [2 lectures]

3. Homomorphisms of rings with 1, isomorphisms, homomorphism of Z into Z_m and the homomorphism f: A[x] ŕ A , f ŕ f(a). Kernel of a homomorphism, ideal, ideal generated by a finite set, principal ideal. Principal ideal domain (PID), theorem: a domain with the Euclidean algoritm (Z, k[x] ) is PID, PID is UFD. Factor ring A/I, construction and universal property. Prime ideals and maximal ideals. Theorem: every proper ideal is contained in some maximal ideal. Theorem: an ideal I is prime (maximal) iff A/I is a domain (a field). [2-3 lectures]

4. Fields have only trivial ideals, every homomorphism of fields is an injection. Prime fields, characteristic of a field. Theorem: f in k[x] is irreducible iff k[x]/(f) is a field. Corollary: the factor ring k[x]/(f) is a field containing the field k, the polynomial f has a root in this field. The definition of the algebraic closure of a field ( without proof of existence and uniqueness). The field of quotients of a domain: construction, examples: from Z to Q, from k[x] to k(x). [ 2 lectures]

5. Group, abelian group, subgroup. Examples: symmetric groups, linear groups, map groups. Cyclic group, the order of an element, the order of a group. Cosets of a subgoup in a group, the index of a subgroup in a group, Lagrange theorem and its applications: every group of prime order is cyclic, Fermat's little theorem. Homomorphism of groups, the kernel of a homomorfphism, normal subgroup, factor group. [2 lectures]

6. Direct product of two groups, the inner characterization of this product. The decompositions of cyclic finite group into a product of cyclic groups with relatively prime orders. Abelian groups: torsion elements, finitely generated torsion free abelian groups, the structure theorem for finitely generated abelian groups (without proof). [1 lecture]

7. The action of a group on a set, the action of a group on itself by left or right translations, Cayley's theorem. Orbits, isotropy groups, fixpoints of an action, free action, efective action. Cardinality of an orbit, of index of the isotropy group. Examples: the action of a symmetric group and the linear group. The action of the symetric group S_n by matrices from GL_n, an application: the sign of a permutation as determinant. The decomposition of a set into orbits, an application: the decomposition of a permutation into disjoint cycles. Automorphisms of groups. The action of a group on itself by inner automorphisms, conjugate classes, the centre of a group as the kernel of the map G ŕ Aut(G). Application: (1) Cauchy's theorem about the existence of an element of prime order, (2) theorem about nontriviality of the centre of any p-group. [3 lectures]

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 notions of groups, rings and fields. and homomorphisms of these structures. Is able to to give and specify different examples of such structures. Knows with proofs basic properties of operations in these structures.

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

3. Know the notions of an action of a group on a set, notions of orbits and stabilizers and relations among them. Is able to describe a relationship among an action of a group on a set and a homomorphism of the group into the group of permutations of the set. Knows applications, in particular: Cayley's theorem the center of a non-trivial p-group is non-trivial and Cauchy's theorem.

4. Knows the notions of normal subgroups and the factor group. Is able to describe normal subroups in distinguished examples of groups and is able to point out subgroups which are not normal. Knows and is able to apply the isomorphism theorem. Knows the notions of the comutant of a group and the abelianization of a group.

5. Knows different types of elements in 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 fundamental constructions of rings and the notions of prime and maximal ideals as well as relationships among them and their characterizations in terms of facto rings.

6. Knows the notions of principal ideals and of principal ideal rings. Is able to describe ideals in the ring of integers, polynomial rings in one indeterminate over fields as well as knows the notion of an euclidean domain and is able to prove that they are principal ideal domains and that the ring of Gauss integers is an euclidean domain.

7. Knows notions of irreducible and prime elements, relationships among them and the definition of unique factorization domains. Is able to sketch a proof of the theorem saying that principal ideal domains are unique factorization domains and knows examples of domains, which are not unique factorization domains. Knows a sketch of Gauss theorem and Eisenstein's criteria and knows how to apply them.

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. Is able to prove that the group of n-th roots of 1 in a field is cyclic and knows a sketch of a construction of an algebraic closure of a field is able to determine cardinalities of finite fields.

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