Topics in algorithmic and computational number theory

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Over the last several decades number theory, that branch of mathematics which deals with the properties of integers, has become a very important area of study in Canada. Indeed, this is one of the most active areas of pure mathematical research in this country. My research involves the discovery of methods, and their computer implementation, to compute certain number theoretic objects. These problems include the seemingly simple problem of determining whether or not an integer is a prime, to more complex problems such as computing the class number of a number field. This latter quantity is of great importance to the corresponding number field; very roughly speaking, it can be considered as a measure of how much the integers of the number field differ in their properties from the integers with which we are all familiar: the whole numbers. My main interest is in finding very efficient ways of solving these kinds of problems.