Algebra II

elective courses

Elements of group theory, field theory, theory of noncommutative rings and modules. Group theory: free groups, solvable groups, semidirect products of groups. Fields: Galois theory and applications. Modules: finitely generated modules over principal ideal domains. Noncommutative rings: matrix algebras, division algebras, Frobenius theorem, algebras of skew polynomials and Weyl algebras.

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.

L. Rowen, Algebra: groups, rings and fields, Wellesley, Massachusetts 1994.

S. Lang, Algebra, Addison-Wesle, 1965.

T.Y. Lam, A First Course in Noncommutative Rings, Springer, 1991.

T.Y. Lam, Exercises in Classical Ring Theory, second edition, Springer, 2003.

1. Knows basic notions of the theory of algebraic field extensions and of solvable groups and is able to use them.

2. Knows the main theorems of Galois theory and their applications to geometric constructions and to solving algebraic equations.

3. Is able to determine the Galois group of a finite field extension and knows how to illustrate the fundamental theorem of Galois theory in characteristic zero.

4. Knows basic notions of the theory of modules over rings and can formulate the structure theorem for finitely generated modules over a PID.

5. Knows main notions and important examples of the theory of noncommutative rings.

6. Knows the constructions of the Hamilton quaternion algebra, Weyl algebra, skew polynomial rings and knows their basic properties.

Points assigned for: homework problems; one midterm test; final written exam

Total numbe rof points: 25 + 50 +125 = 200

Final result based on the total number of points