HIGHER RECIPROCITY LAWAND APPLICATIONS

• Main Author: Sean Hendarto, William
• Format: Final Project
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/83453
• Institution: Institut Teknologi Bandung

The thesis introduces some of the statements and results available in algebraic number theory as well as their applications using a variety of different techniques. The main subject of interest is prime ideal factorization of a number field with their applications. More specifically, we will look at the Galois group of a number field and connect those group with the prime ideal factorization of its ring of integers. This is what is meant from the term "reciprocity law" and only abelian extensions are considered. The first part focuses on the properties of cyclotomic fields (fields generated by e2?i n ) through the so called cyclotomic reciprocity law, and further properties of such fields through the Kronecker-Weber theorem and Dirichlet’s theorem in arithmetic progressions. Both theorem require results from the geometry of the ring of integers that will only be mentioned briefly. The second part uses the main theorems in Class field theory, which gives a comprehensive property of abelian extensions in both algebraic terms and in analytic terms, and the j-invariant to construct a polynomial H?4m(t) which characterizes the set of primes p representable as x2+my2 with x,y,m ? N. The last part gives a brief introduction on the Kronecker Jugendtraum using elliptic curves with complex multiplication and how such torsion points combined with the jinvariant j(z) generate the ray class field of K.