250060 VO Algebraic number theory (2019W)

One of the major sources for the development of algebraic number theory was the
attempt to generalize the quadratic reciprocity law to higher
power residues. While the quadratic reciprocity law can be formulated entirely in Q, i.e. it
can be formulated using only rational
numbers the formulation of higher reciprocity laws involves n-th roots of unity, i.e. it
necessarily takes place in a finite extension of Q. This creates the need to consider
finite (i.e. algebraic) extensions of Q - the so called
algebraic number fields - in analogy with Q. The goal of algebraic number theory thus is
to extend the essential properties of
the field Q to finite extensions of Q. Briefly, this means
- to find an analogoue in number fields of the ring of integers Z in Q; in particular this analogoue is a ring and it should have some kind of factorization into primes property.
The study of these rings is a fundament of algebraic number theory.
- to find an analogoue of the embedding of Q into R (and also into p-adic fields Q_p); this makes possible to relate algebraic numbers to geometry and analysis.
If time permits we can occasionally touch on aspects of the modern formulation of
algebraic number theory which evolved in connection
with the geometric interpretation of algebraic numbers.
Prerequisites for the course ``Algebraic number theory'' are Algebra 1 and Algebra 2.