We present the basics of number theory emphasizing the striking similarity between the properties of usual integers and those of polynomials over a finite field. This leads to understanding that number theory and algebraic geometry are, in fact, the same domain. The core of number theory is an elementary problem, so we stick to concrete examples.
Prerequisites
It is assumed that students are familiar with the concepts of
- homomorphisms
- groups, subgroups, cosets, normal subgroups, quotient groups, conjugate classes, group actions
- rings, subrings, ideals, principal ideals, quotient rings
- fields, rings of polynomials, Euclidean algorithm, Bézout’s theorem
Program of the course
- Finite fields.
Prime fields. General construction of finite fields. Cyclicity of the multiplicative group. Frobenius map. The Legendre symbol. Gaussian sums. Gauss’ reciprocity law. - p-adic numbers.
Projective limit construction of p-adic numbers. Absolute values on k(T) and on Q. Ostrowski’s theorem. Topological construction of p-adic numbers. Hensel’s lemma. Multiplicative group of Q_p. The Teichmuller map. Power series. p-adic logarithm. - Galois theory.
Constructions of algebraic extensions. Algebraic closure. Homomorphisms of field extensions. Normal and separable extensions. Galois group. Main theorem. Basic examples. the Fundamental theorem of algebra. Cyclic extensions. Cyclotomic fields. Norm and trace. Decomposition of the tensor product. - Algebraic number fields.
Ring of integers. Action of the complex conjugation on homomorphisms. Geometry of numbers. Minkowski’s theorem. Dirichlet’s units theorem. Classes of ideals. the finiteness theorem. Quadratic fields. - Extensions of global fields.
Discrete valuation rings. Dedekind rings. Group of fractional ideals. Localisation. Extensions of Dedekind rings. Lattices. Dual lattice. Different and discriminant. Global and local fields. Decomposition of prime ideals in finite extensions. Lifting of absolute values to the finite extensions of local and global fields. Ramification degree and inertia index. Unramified extensions. Totally ramified extensions.
If time permits (normally it does), we are also going to study the foundations of p-adic calculus (Krasner’s lemma, Tate field, power series and the Newton polygon).
Grading & Assessment
The students will receive the lecture notes after each lecture.
Approximately 50 homework problems will be suggested during the semester (ranging from basic to average). There will also be two tests (a midterm and a final). The final grade will be close to the average of the best two out of three marks (homework, midterm test, and final test). In borderline cases, the third mark will be taken into account.
Textbooks
- S. Lang, Algebraic number theory, Springer, 1994.
- S. Lang, Algebra, Springer, 2002.
- J.-P. Serre, A course in arithmetic, Springer, 1996.
- Z.I. Borevich, I.R. Shafarevich, Number Theory, Academic Press, 1966.