Powers of the fundamental ideal in the Witt ring (Q5942789)

Using the results of Voevodsky and Orlov-Vishik-Voevodsky on the Milnor conjectures, the authors prove several new theorems about the Witt ring \(W= W(K)\) of a field \(K\) of characteristic different from 2. In particular they show: NEWLINENEWLINENEWLINETheorem 1.5: Let \(I^n\) be the \(n\)th power of the fundamental ideal \(I\subset W\), let \(J_n\) be the ideal of all elements \(\varphi\) of Knebusch degree \(\geq n\). Then NEWLINE\[NEWLINEJ_n= I^n \quad\text{(for all }n).NEWLINE\]NEWLINE Theorem 2.2: Let \(\omega\) be a \(k\)-fold Pfister form. Then NEWLINE\[NEWLINEI^{n+k}\cap W\omega= I^n\omega.NEWLINE\]NEWLINE Theorem 2.3: Let \(I_\omega= \text{ann}(\omega)\) be the annihilator ideal of \(\omega\), which is generated by binary forms. Then NEWLINE\[NEWLINEI^n\cap I_\omega= I^{n-1}\cdot I_\omega.NEWLINE\]NEWLINE Corollary 2.7: Let \(T\) be a preordering of \(K\), let \(I\) be the ideal of \(W\) generated by the elements \(\langle 1,-t\rangle\) with \(t\in T\). Assume \(\varphi\in I^n\), \(\text{sign}_P \varphi=0\) for every ordering \(P\) of \(K\) containing \(T\). Then NEWLINE\[NEWLINE\varphi\in I^{n-1} I_T.NEWLINE\]NEWLINE It is mentioned, but not proved in detail, that the above results also lead to a proof of Lam's ``Open Problem B''. The idea is as follows: NEWLINENEWLINENEWLINELet \(\varphi\in W\) be a form with \(2^n\mid\operatorname {sign}\varphi\). By the Normality Theorem, there is a large number \(m\) such that \(\langle 1,1\rangle^{m-n}\cdot \varphi= \chi\in I^m\). Theorem 2.2 (with \(\chi\) instead of \(\varphi\) and \(\omega= \langle 1,1\rangle^{m,n})\) shows that \(\chi= \omega\cdot \psi\) with \(\psi\in I^n\), hence \(\varphi\in I^n+ W_{\text{tor}}\). NEWLINENEWLINENEWLINEThis proof is much better than the previous (partial) proofs of Dickmann-Miraglia and Monnier. NEWLINENEWLINENEWLINEThe final \S 3 contains an explicit presentation of \(I^n\) by generators and defining relations (Theorem 3.1) as well as some further applications related to the stability index \(\text{st}(K)\).