Sums of fourth powers of real algebraic functions (Q1331749)

Let \(K\) be a formally real field and \(P_ n(K)\) its \(n\)-th Pythagoras number. The author proves the following bound for \(P_ 4(K)\): If \(P_ 2(K)=2\) and \(3\in K^ 2\), then \(P_ 4(K)\leq 6\). In the proof of this theorem he uses results on the real holomorphy ring \(H(K)\) of \(K\) and the following lemma: Let \(a\in H(K)^ \times\) be a sum of two squares in \(K\), then there are \(u,v\in H(K)^ \times \cap \Sigma K^ 2\) such that \(a= u^ 2+ v^ 2\).