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Sur deux invariants d’un corps de niveau fini. (French) Zbl 0417.12005
The author considers fields \(K\) in which \(-1\) is a finite sum of squares. If \(q=\text{card}(K^*/{K^*}^2)\) and \(s\) is a minimal number for which \(-1 = \alpha_1^2+\ldots+\alpha_n^2\) then \(s =2^k\) for some \(k\) and \(q\), \(s\) and \(k\) are connected by a few known relations.
Main construction. In the previous notation: \(\alpha_0^2+\alpha_1^2+\ldots +\alpha_s^2=0\) where \(\alpha_0= 1\) and obviously all \(\alpha_i\neq 0\). Let \(S= \{0,1,\ldots,s)\) and \(\mathcal P_i(S)\) be the set of all parts of \(S\) consisting of \(i\) elements. Let further \(\Gamma_{s,i}\) be the graph in which elements of \(\mathcal P_i(S)\) are vertices and \((I,J)\) for \(I,J\in\mathcal P_i(s)\) is an edge if and only if \(\text{card}\, (I-J) = 1\). One proves that if \(T(s,i)\) is the length of minimal decomposition of \(\Gamma_{s,i}\) in cliques then \[ q_{i+1}(K) - q_i(K)\geq T(s,i+1) \] where \[ q_i(K) = \text{card}\biggl(\biggl\{\sum_{k=1}^s x_k^2\mid x_1,\ldots,x_s\in K\biggr\}\biggr)/{K^*}^2\right). \] Besides \(T(s,i)=T(s,s+1-i)\) and \(T(s,2)=s-1\) for \(s\geq 2\). One describes in detail the graphs \(\Gamma_{1,i}\), \(\Gamma_{2,i}\) and in the case \(K=\mathbb Q_2\) the graph \(\Gamma_{4,i}\).

MSC:
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
11E04 Quadratic forms over general fields
05C99 Graph theory
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References:
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