Before we take up specific examples, we would like to say something about how representations of an integer as a sum of squares (SOS) in Q is related to that in Z, and indeed more generally about how the representation of an element a of an integral domain A as an SOS in A is related to the representation of a as an SOS in the field of quotients F of A. We paraphrase our questions more explicitly as follows:

Question 1. If a ∈ A is an SOS of n elements of F then is a an SOS of elements of A?

Question 2. If the answer to Question 1 is ‘yes’ then is a an SOS of the same number n of elements of A?

The lemma of Cassels proved in the last chapter is an excellent example of a problem of this nature, where the answer to both the above questions is given in the affirmative (A being the ring K[t], K a field). Note that the answer to the first question was proved to be in the affirmative by Artin already in 1927; but the second was not answered then.

Another very instructive example is provided in the case A = Z, so that F = Q; the answers to both the questions being in the affirmative and this result is often called the Davenport-Cassels lemma.