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CLASS NUMBERS OF NUMBER RINGS : EXISTENCE AND COMPUTATIONS

This paper gives a brief introduction to the theory of algebraic numbers, with an eye toward computing the class numbers of certain number rings. The author adapts the treatments of Madhav Nori (from [1]) and Daniel Marcus (from [2]). None of the following work is original, but the author does include some independent solutions of exercises given by Nori or Marcus. We assume familiarity with the basic theory of fields. 1. A Note on Notation Throughout, unless explicitly stated, R is assumed to be a commutative integral domain. As is customary, we write Frac(R) to denote the field of fractions of a domain. By Z, Q,R,C we denote the integers, rationals, reals, and complex numbers, respectively. For R a ring, R[x], read ‘R adjoin x’, is the polynomial ring in x with coefficients in R. 2. Integral Extensions In his lectures, Madhav Nori said algebraic number theory is misleadingly named. It is not the application of modern algebraic tools to number theory; it is the study of algebraic numbers. We will mostly, therefore, concern ourselves solely with number fields, that is, finite (hence algebraic) extensions of Q. We recall from our experience with linear algebra that much can be gained by slightly loosening the requirements on the structures in question: by studying modules over a ring one comes across entire phenomena that are otherwise completely obscured when one studies vector spaces over a field. Similarly, here we adapt our understanding of algebraic extensions of a field to suitable extensions of rings, called ‘integral.’ Whereas Z is a ‘very nice’ ring—in some sense it is the nicest possible ring that is not a field—the integral extensions of Z, called ‘number rings,’ have surprisingly less structure. Arbitrary number rings, for example, do not even have unique factorization into irreducibles—let alone the property of every ideal being principal. As in other subjects, when we are trying to understand objects of a certain type, we try to group them into types. A natural ‘class’ of number rings would be the ones that happen to be PIDs. We will define the ‘class number’ of a ring to measure, in some sense, how badly a certain ring fails to be a PID. We will then show that every number ring has finite class number. In other words, every number ring is ‘somewhat close’ to being a PID. Finally, we will explicitly calculate the class number of some number rings. This might require some slickness, because in general there is no good way to calculate the class number explicitly.