Algebra I 1000-113bAG1b

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]