Given g1;:::;gs 2 R(X) = R(X1;:::;Xn) such that the semialgebraic set K := fx 2 Rn j gi(x) 0 for all ig is compact. Schmudgen's Theorem says that if f 2 R(X) such that f > 0 on K, then f is in the preordering in R(X) generated by the gi's, i.e., f can be written as a nite sum of elements g e 1 1 :::ge s s , where is a sum of squares in R(X) and each ei2f0; 1g. Putinar's Theorem says that under a condition stronger than compactness, any f > 0 on K can be written f = 0 + 1g1 + + sgs, where i 2 R(X). Both of these theorems can be viewed as statements about the existence of certicates of positivity on compact semialgebraic sets. In this note we show that if the dening polynomials g1;:::;gs and polynomial f have coecients in Q, then in Schmudgen's Theorem we can nd a representation in which the 's are sums of squares of polynomials over Q. We prove a similar result for Putinar's Theorem assuming that the set of generators contains N P X2 i for some N2 N.