Let $F$ be a field in which we have elements satisfying $a^2+b^2+c^2 = −1$. Show that there exist elements satisfying $d^2+e^2 = −1$.

Question

Let $F$ be a field in which we have elements $a, b$, and $c$ satisfying $a^2+b^2+c^2 = −1$. Show that there exist elements $d$ and $e$ of $F$, satisfying $d^2+e^2 = −1$.

Any hint?

This is an excercise from the book: The Linear Algebra a Beginning Graduate Student Should Know; Golan

Answer 1

Hint $\ $ Let $\ d=1\ $ in $\,(a^2+b^2)(\overbrace{a^2+b^2+c^2+d^2}^{\large =\, 0})\, =\, (\overbrace{a^2+b^2}^{\large x})^2 + (\overbrace{ac-bd}^{\large y})^2 + (\overbrace{ad+bc}^{\large z})^2$

That yields $\ x^2+y^2+z^2 = 0\ $ which, divided by $\,x^2,\,$ yields the result.

Remark $\ $ The latter two summands arise from the Brahmagupta–Fibonacci identity