In this thesis we investigate the integral trace form of cyclotomic fields. We first show that the formula in the general case, that is, cyclotomic fields of any conductor, can be reduced to the case where the conductor is square-free. As corollaries, we obtain: 1) a symmetric polynomial representation of the form in cyclotomic fields of prime conductor (a result earlier obtained by Interlando) and 2) the form in the case of cyclotomic fields of prime power conductor. We then use the latter form to produce a succinct algorithm for computing the nonzero minimum of the form within a certain submodule of the ring of integers of the cyclotomic field. One of the applications of this is in finding the shortest (nonzero) vector of a lattice, a classical problem in geometry of numbers.
