Witt rings of global function fields. (Q5933549)

\textit{F. DeMeyer}, \textit{D. Harrison} and \textit{R. Miranda} [Mem. Am. Math. Soc. 394, 1--42 (1989; Zbl 0668.10028)] described the Witt ring \(W({\mathbb Q})\) of the rational number field \(\mathbb Q\) by defining the ring structure on an explicitly constructed set \(V(\mathbb Q)\) and showing that the rings \(W({\mathbb Q})\) and \(V(\mathbb Q)\) are isomorphic. In the present paper a similar description is given for the Witt ring of any global function field \(K\). The structure of \(W(K)\) depends on the level \(s(K)\) of \(K\). If \(s(K) = 1\), the autor takes \(V(K) = {\mathbb Z}/2{\mathbb Z} \times {\dot K}/{\dot K}^2 \times \Gamma_K\), where \(\Gamma_K\) is the group of finite even-order subsets of the set of all primes of \(K\) with symmetric difference as operation. The addition in \(V(K)\) is componentwise except in the third component, \((n,a,A)+(m,b,B)=(n+m,ab,A+B+(a| b))\), where \((a| b)\) is the set of primes \(P\) at which the Hilbert symbol \((a,b)_P\) assumes the value \(-1\). Multiplication is defined by the formula NEWLINE\[NEWLINE(n,a,A) \cdot (m,b,B)= (nm,a^mb^n,mA+nB+(mn+1)(a| b)).NEWLINE\]NEWLINE The author proves that \(V(K)\) is a ring isomorphic to the Witt ring \(W(K)\). A similar description of \(W(K)\) is given when \(s(K)=2\).NEWLINENEWLINEThe results of this paper have been generalized to all global fields by the author and \textit{A.~Sładek} [Commun. Algebra 31, No. 7, 3195--3205 (2003; Zbl 1069.11012)].