The Qth-power algorithm produces a useful canonical P-module presentation for the integral closures of certain integral extensions of $P:=mathbf{F}[x_n,...,x_1]$, a polyonomial ring over the finite field $mathbf{F}:=mathbf{Z}_q$ of $q$ elements. Here it is shown how to use this for several small primes $q$ to reconstruct similar integral closures over the rationals $mathbf{Q}$ using the Chinese remainder theorem to piece together presentations in different positive characteristics, and the extended Euclidean algorithm to reconstruct rational fractions to lift these to presentations over $mathbf{Q}$.
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