# If -1 is a sum of squares, then any element is a sum of squares

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?

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