Math-GA 2130.001: Algebra I, Thu. 7:10pm, WWH 202

Instructor: Yuri Tschinkel

Office hours:

• Thu 4:30-5:30pm or by appointment

Syllabus:

• Basic notions of algebra: groups, rings, and fields
• Symmetric groups, linear groups
• Structure theorems, Sylow subgroups
• Algebraic number fields, finite fields, field extensions
• Galois theory
• Polynomial rings, unique factorization
• Geometry of Numbers, ideals, ideal class groups, units
• Rings, algebraic integers, ideals

Grading:

• Homework: 20%
• Midterm October 22: 30% / Solutions
• Final December 10: 50%

Recommended books:

• M. Artin, Algebra, Pearson Education
• A. Chambert-Loir, A Field guide to Algebra, Springer

Lectures:

1. Arithmetic in Z and C[x], Euclidean algorithm, Euler function, congruences, Wilson's theorem, Fermat's little theorem, primitive roots, Artin's root conjecture
2. Applications of congruences, equations modulo p, theorem of Chevalley-Warning, Legendre symbol, Gauss sums, quadratic reciprocity, Hasse-Minkowski principle for quadratic forms in 3 variables, Reichard's counterexample
3. Finite groups: elementary properties, subgroups, normal subgroups, index, group actions, orbits, stabilizer, centralizer, Cauchy's theorem
4. Sylow subgroups, Sylow theorems, applications, solvable groups, symmetric group, transpositions
5. Introduction to invariant theory for finite group actions: elementary symmetric polynomials, computing invariants, Reynolds operator
6. Field extensions, field isomorphisms, Galois theory
7. Algebraic integers, ideals

Homework:

1. Elementary number theory, (due September 17) / Solutions
2. Legendre symbol, quadratic reciprocity, (due September 24) / Solutions
3. Basic group theory, (due October 1) / Solutions
4. Invariant theory, (due October 8) / Solutions
5. Field extensions, (due October 15) / Solutions
6. Algebraic integers, ideals, (due November 5) / Solutions
7. Algebraic integers, (due November 12) / Solutions
8. Splitting of ideals, ideal class groups, (due November 19) / Solutions