Let $K$ be a finite field with $p^2$ elements. Show that $3$ is square in $K$.
I know that 3 is sum of two squares.
Thanks.
2026-03-25 09:36:38.1774431398
3 is Square in finite field
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Hint: Observe that the splitting field of the polynomial $x^2-3$ over $\mathbb{F}_p$ is either $\mathbb{F}_p$ or $\mathbb{F}_{p^2}$, so that either way, the square roots of $3$ lie in $\mathbb{F}_{p^2}$.