Math 80220 Topics in Algebra 2 (Spring
2014)
Introduction to Algebraic Number Theory
| Date | Topics | Source | Sage Log |
| 2014-01-15 | Course overview | Overview | |
| 2014-01-17 | Fields, extensions, degrees, primitive elements, algebraic extensions and elements. Number rings and rings of integers of number fields. | Lecture 2 | Sage session |
| 2014-01-20 | Algebraic integers, traces and norms. | Lecture 3 | Sage session |
| 2014-01-22 | Discriminants, integral bases. | Lecture 4 | |
| 2014-01-24 | Rings of integers of cyclotomic fields, Dedekind domains | Lecture 5 | Sage session |
| 2014-01-27 | Fractional ideals in Dedekind domains are invertible | Lecture 6 | Sage session |
| 2014-01-29 | Unique factorization in Dedekind domains, the Chinese Remainder Theorem and ideals generated by two elements. | Lecture 7 | Sage session |
| 2014-01-31 | Unique factorization domains, prime ideals under extensions. | Lecture 8 | Sage session |
| 2014-02-03 | Ideals under extensions, norms of ideals. | Lecture 9 | Sage session |
| 2014-02-05 | Norms of ideals, ramification and inertia indices. | Lecture 10 | (Includes code not run in class) Sage session |
| 2014-02-07 | Splitting of ideals, ramification and Galois theory | Lecture 11 | Sage session |
| 2014-02-10 | Galois theory and prime ideals | Lecture 12 | (Includes code not run in class) Sage session |
| 2014-02-12 | Subfields fixed by the decomposition and inertia groups | Lecture 13 | |
| 2014-02-14 | Quadratic reciprocity, higher ramification and the different | Lecture 14 | |
| 2014-02-17 | Ramification and the discriminant; finitess of the class group | Lecture 15 | |
| 2014-02-21 | Finiteness of the class group and the geometry of numbers 1 | Lecture 16 | |
| 2014-02-24 | Finiteness of the class group and the geometry of numbers 2 | Lecture 17 | Sage session |
| 2014-02-26 | The Dirichlet unit theorem 1 | Lecture 18 | Sage session |
| 2014-02-28 | The Dirichlet unit theorem 2 | Lecture 19 | |
| 2014-03-03 | Counting ideals 1 | Lecture 20 | Sage session |
| 2014-03-05 | Counting ideals 2; Dirichlet series | Lecture 21 | Sage session |
| 2014-03-07 | The analytic class number formula | Lecture 22 | |
| 2014-03-17 | The Dedekind zeta function at s=0; characters and L-functions | Lecture 23 | |
| 2014-03-19 | The Dedekind zeta function and L-functions of characters | Lecture 24 | |
| 2014-03-21 | Dirichlet's theorem on primes in arithmetic progressions and the Chebotarev density theorem | Lecture 25 | |
| 2014-03-24 | Applications of Chebotarev; The values of L-functions at negative integers | Lecture 26 | |
| 2014-03-26 | Gauss sums and the values of L-functions at 1 | Lecture 27 | |
| 2014-03-28 | The conductor-discriminant formula | Lecture 28 | |
| 2014-03-31 | Cyclotomic units 1 | Lecture 29 | |
| 2014-04-02 | Cyclotomic units 2 | Lecture 30 | |
| 2014-04-04 | A peculiar integral | Lecture 31 | |
| 2014-04-07 | Varieties, curves, local rings at smooth points, uniformizers. | Lecture 32 | |
| 2014-04-09 | Morphisms of curves, ramification, Frobenius, separable extensions of function fields. | Lecture 33 | |
| 2014-04-11 | Rings of integers in function fields: prime ideals, unique factorization | Lecture 34 | |
| 2014-04-14 | Rings of integers in function fields as Dedekind domains; divisors and Riemann-Roch | ||
| 2014-04-16 | Riemann-Roch and rings of integers; Weierstrass equations for elliptic curves. | Lecture 36 | |
| 2014-04-23 | L-functions on function fields and their functional equation 1 | Lecture 37 | |
| 2014-04-25 | L-functions on function fields and their functional equation 2 | Lecture 38 | |
| 2014-04-28 | Irreducible polynomials mod p in arithmetic progressions; elliptic curves are abelian groups | Lecture 39 | |
| 2014-04-30 | Elliptic curves over finite fields | Lecture 40 |
