MA 213: Algebra II
Credits: 3:1
Prerequisite courses: MA 212
Part A: Field theory
- Theory of symmetric polynomials – Newton’s theorem
- Basic theory of field extensions
- Algebraic and transcendental extensions (and transcendence degree)
- Construction with straight edge and compass; Gauss-Wantzel theorem
- Algebraic closure – Steinitz’s theorem
- Splitting fields, normal extensions
- Separable extensions
- Finite fields: construction, subfields, Frobenius
- Primitive element theorem
- Dedekind-Artin linear independence of (semi)group characters
Part B: Galois theory
- Fundamental theorem of Galois theory (including Normal Basis Theorem)
- Composite extensions and Galois group
- Galois group of cyclotomic extensions, finite fields
- Galois groups of polynomials, Fundamental theorem of Algebra
- Solvable and radical extensions, insolvability of a quintic
Suggested books and references:
- Artin, M., Algebra, Prentice Hall of India, 1994.
- Dummit, D. S. and Foote, R. M., Abstract Algebra, McGraw-Hill, 1986.
- Lang, S., Algebra (3rd Ed.), Springer, 2002.
- Jonathan Alperin and Rowen Bell, Groups and Representations, Graduate Texts in Mathematics 162, Springer Verlag, 1995.
- Hungerford, Algebra, Graduate Texts in Mathematics 73, Springer Verlag, 1974.
- Galois Theory, Artin, E., University of Notre Dame Press, 1944.
- Nathan Jacobson, Basic Algebra I & II, Dover, 2009.
- Nathan Jacobson, Lectures in Abstract Algebra I, II & III, Graduate Text in Mathematics, Springer Verlag, 1951.