Quadratic Number Fields
In this senior seminar we will explore the algebra and arithmetic of quadratic extensions of the rational numbers. We will study the rings of integers in these extensions, the structure of the unit group in these rings and unique factorization of ideals in Dedekind domains. We will investigate fractional ideals, splitting of primes, the class group and the finiteness of the class number. Some of the ideas and topics introduced are methods, p-adic methods, cyclotomic theory, Dirichlet’s Units Theorem, Quadratic and Biquadratic Reciprocity and quadratic forms. Using these ideas as a springboard students will investigate a topic of their choosing and write their thesis.