### MA 213: Algebra II

#### Part A: Field theory

1. Theory of symmetric polynomials – Newton’s theorem
2. Basic theory of field extensions
3. Algebraic and transcendental extensions (and transcendence degree)
4. Construction with straight edge and compass; Gauss-Wantzel theorem
5. Algebraic closure – Steinitz’s theorem
6. Splitting fields, normal extensions
7. Separable extensions
8. Finite fields: construction, subfields, Frobenius
9. Primitive element theorem
10. Dedekind-Artin linear independence of (semi)group characters

#### Part B: Galois theory

1. Fundamental theorem of Galois theory (including Normal Basis Theorem)
2. Composite extensions and Galois group
3. Galois group of cyclotomic extensions, finite fields
4. Galois groups of polynomials, Fundamental theorem of Algebra
5. Solvable and radical extensions, insolvability of a quintic

#### Suggested books and references:

1. Artin, M., Algebra, Prentice Hall of India, 1994.
2. Dummit, D. S. and Foote, R. M., Abstract Algebra, McGraw-Hill, 1986.
3. Lang, S., Algebra (3rd Ed.), Springer, 2002.
4. Jonathan Alperin and Rowen Bell, Groups and Representations, Graduate Texts in Mathematics 162, Springer Verlag, 1995.
5. Hungerford, Algebra, Graduate Texts in Mathematics 73, Springer Verlag, 1974.
6. Galois Theory, Artin, E., University of Notre Dame Press, 1944.
7. Nathan Jacobson, Basic Algebra I & II, Dover, 2009.
8. Nathan Jacobson, Lectures in Abstract Algebra I, II & III, Graduate Text in Mathematics, Springer Verlag, 1951.

#### All Courses

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 20 Mar 2023