Algebra II 1000-134AG2

1. Elements of group theory.
i) Free groups, presentations of groups (groups defined by generators and relations).
ii) Semidirect product of groups. Exact sequences, split exact sequences.
iii) Solvable groups; derived subgroup, solvability of permutation groups S_n, for n<5.
iv) Simple groups; simplicity of groups A_n, for n>4.
2. Elements of field theory.
i) Field extensions, groups of automorphisms. Extension by a root of a polynomial, splitting field of a polynomial, normal extensions and their universal property.
Algebraic extensions, algebraic closure - construction and uniqueness.
Roots of unity. Fields of p^n elemets (existence).
ii) Galois theory of finite field extensions in characteristic 0.
Irreducible polynomials in characterictic 0 have no multiple roots. Theorem of Abel, Galois extensions. Main Theorem of Galois Theory.
ii) Applications of Galois theory: the Fundamental Theorem of Algebra, solvable extensions, solving equations by radicals (1-2 lectures)
iii) Applications of Galois theory.
Geometric constructions (constructability implies that the degree of the extension is a power of 2).
Extensions solvable by radicals.
3. Elements of the theory of modules.
Modules, direct sum, finitely generated modules, torsion elements.
Homomorphisms of modules, the kermel, factor module, exact sequences of modules, splittings.
Free modules.
Classification of finitely generated modules over PID's. Corollaries: classification of finitely generated abelian groups, Jordan's Theorem from linear algebra on canonical form of matrices.
4. Elements of noncommutative rings.
i) Examples: endomorphisms rings of vecor spaces, matrix rings, one-sided ideals, simple rings.
ii) Division rings, Quaternion algebra. Frobenius theorem on finite dimensional division algebras over R.
iii) Weyl algebra (in characteristic 0) - definition and interpretation in terms of differential operator algebra and in terms of skew polynomials. This algebra is a domain and it is simple.