Higher product levels of domains

We give an explicit bound for the n-th Pythagorean number of a simple extension Q(d)/Q in Section 2 and prove a noncommutative version of Hilbert identities in Section 3. These two results are used in Section 4 in the proof of our main result. In section 5, we show that the n-th product level of an Ore domain is equal to the n-th product level of its skew field of fractions.