The Ideal Class Group

We present a concise and self-contained definition of the ideal class group, which is useful for proving facts about zero sets of Diophantine equations, and discuss a few relevant key facts. We approach this by first assembling some preliminary definitions regarding algebraic integers, and subsequently delving into several useful results about lattices, including Minkowski’s lemma. Then, returning to our goal, we construct the ideal class group for the ring of integers in an imaginary quadratic field. Finally, applying the facts we have accumulated, we prove that the group is always finite and analyze a few examples. 1 Background and Definitions Below, we assemble some definitions which will help us achieve our two initial goals of describing the prime factorization of ideals and constructing the ideal class group, and inform our examples further below. The notion of an imaginary quadratic number field underlies the structure of the ideal class group. 1.1 Imaginary Quadratic Number Fields Definition 1. We define a field F as being an imaginary quadratic number field if F = Q[ √ d], where Q[ √ d] is the field consisting of all complex numbers of the form a+ b √ d such that a, b ∈ Q and the constant d ∈ Z satisfies d < 0. In general, only square-free values of d are of interest; otherwise, we can easily factor the square out of d and include the square’s root in b. 1.2 The Ring of Integers in Q[ √ d] Definition 2. We call an element δ ∈ C an algebraic integer if it is in the zero set of a monic polynomial f(x) ∈ Z[x].