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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 $$\Gamma_{4,i}$$.
Show Scanned Page ##### MSC:
 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) 11E04 Quadratic forms over general fields 05C99 Graph theory
##### Keywords:
graph; square classes; formally nonreal field; finite level
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##### References:
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