We have a field $F$ of characteristic different from $2$ in which there exist $a_1, \dots, a_n$ such that $$ a_1^2 + \dots + a_n^2 = -1. $$

Prove that for any $c \in F$, there exist $b_1, \dots, b_k$ such that $$ b_1^2 + \dots + b_k^2 = c. $$

I have no idea where to start from. Would anybody give me a clue?


1 Answer 1


$$c=\left(\frac{c+1}2\right)^2-\left(\frac{c-1}2\right)^2 =\left(\frac{c+1}2\right)^2+(a_1^2+\cdots+a_n^2)\left(\frac{c-1}2\right)^2.$$

  • $\begingroup$ Could not be more clear.. Thanks! $\endgroup$
    – Tom
    Sep 10, 2019 at 6:54

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