  Cochran defined the nth-order integral Alexander module of a knot in the
three sphere as the first homology group of the knot's (n+1)th-iterated abelian
cover. The case n=0 gives the classical Alexander module (and polynomial).
After a localization, one can get a finitely presented module over a principal
ideal domain, from which one can extract a higher-order Alexander polynomial.
We present an algorithm to compute the first-order Alexander module for any
knot. As applications, we show that these higher-order Alexander polynomials
provide a better bound on the knot genus than does the classical Alexander
polynomial, and that they detect mutation. Included in this algorithm is a
solution to the word problem in finitely presented Z[Z]-modules.
