ALGEBRAIC NUMBER THEORY
Algebraic preliminaries: Algebraic field extensions: Normal, separable and Galois extensions. Euclidean rings, principal ideal domains and factorial rings. Quadratic number fields. Cyclotomic number fields. Algebraic integers: Integral extensions: Algebraic number fields and algebraic integers. Norms and traces. Resultants and discriminants. Integral bases. Class numbers:Lattices and Minkowski theory. Finiteness of class number. Dirichlet’s unit theorem. Ramification Theory: Discriminants.
Books
- Artin, E., Galois Theory, University of Notre Dame Press, 1944
- Borevich, Z. and Shafarevich, I., Number Theory, Academic Press, New York, 1966.
- Cassels, J.W. and Frohlich, A., Algebraic Number Theory, Academic Press, New York, 1948
- Hasse, H., Zahlentheorie, Akademie Verlag, Berlin, 1949.
- Hecke, E., Vorlesungen uber die Theorie der algebraischen Zahlen, Chelsea, New York, 1948.
- Samuel, P., Algebraic Theory of Numbers, Hermann, 1970
PREREQUISITES:
- Linear algebra (MA 219 or equivalent)
- Basic algebra: Groups, rings, modules (MA 212 or equivalent), and algebraic field extensions