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 is in fact one and the same domain. The core of number theory being an elementary problem, we stick to concrete examples.
Prerequisites: It is supposed that the students are familiar to the concepts of
- group, subgroup, coset, normal subgroup, quotient group, conjugate class, action of the group
- ring, subring, ideal, principal ideal, quotient ring
- field, ring of polynomials, euclidean algorithm, Bezout’s theorem
- 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 theorem. Topological construction of P-adic numbers. Henzel’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 theorem. Dirichlet’s units theorem. Classes of ideals. Finiteness theorem. Quadratic fields.
- Extensions of global fields.
Discretely valuated 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.
- 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.