4400intro - FOUNDATIONS AND THE FUNDAMENTAL THEOREM PETE L...

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FOUNDATIONS AND THE FUNDAMENTAL THEOREM PETE L. CLARK 1. Foundations What is number theory? This is a difficult question to answer: number theory is an area, or collection of areas, of pure mathematics that have been studied for well over two thousand years. As such, it means different things to different mathematicians. Nevertheless the question is not nearly as subjective as “What is truth?” or “What is beauty?”: all of the things that various people call number theory are related, in fact deeply and increasingly related over time. If you think about it, it is hard to give a satisfactory definition of any area of mathematics that would make much sense to someone who has not taken one or several courses in it. One might say that analysis is the study of limiting processes, especially summation, differentiation and integration; that algebra is the study of algebraic structures like groups, rings and fields; and that topology is the study of topological spaces and continuous maps between them. But these descriptions function more by way of dramatis personae than actual explanations; less preten- tiously, they indicate (some of) the objects studied in each of these fields, but they do not really tell us which properties of these objects are of most interest and which questions we are trying to answer about them. Such motivation is hard to provide in the abstract – much easier, and more fruitful, is to give examples of the types of problems that mathematicians in these areas are or were working on. For instance, in algebra one can point to the classification of finite simple groups, and in topology the Poincar´ e conjecture. Both of these are problems that had been open for long periods of time and have been solved relatively recently, so one may reasonably infer that these topics have been central to their respective subjects for some time. What are the “objects” of number theory analogous to the above description? A good one sentence answer is that number theory is the study of the integers, i.e., the positive and negative whole numbers. Of course this is not really satisfactory: astrology, accounting and computer sci- ence, for instance, could plausibly be described in the same way. What properties of the integers are we interested in? The most succinct response seems to be that we are interested in the integers as a ring : namely, as endowed with the two fundamental operations of addition + and multiplication · and – especially – the interactions between these two operations. 1
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2 PETE L. CLARK Let us elaborate. Consider first the non-negative integers – which, as is tradi- tional, we will denote by N – endowed with the operation +. This is a very simple structure: we start with 0, the additive identity, and get every positive integer by repeatedly adding 1.
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