Isotropy and factorization in reduced Witt rings (Q5949807)

Let \(R\) be a reduced Witt ring. A form \(q\in R\) is called totally indefinite if \(|\text{sgn}_\alpha q|<\dim q\) for all orderings \(\alpha\) of \(R.\) Of course, an isotropic form is totally indefinite, but it may happen that a totally indefinite form is anisotropic. The main result of the paper says that if the chain length \(cl(R)\) of \(R\) is finite and \(q\in R\) is anisotropic, then \(\dim q\leq \frac{1}{2}cl(R)\max\{|\text{sgn}_\alpha q|^2: \alpha\in X(R)\}.\) This allows the author to prove that \(R\) satisfies the ascending chain condition on principal ideals iff \(cl(R)<\infty\) and to get some new results on the factorization of forms into products of irreducible forms.