## Each Element in a Finite Field is the Sum of Two Squares

## Problem 511

Let $F$ be a finite field.

Prove that each element in the field $F$ is the sum of two squares in $F$.

Let $F$ be a finite field.

Prove that each element in the field $F$ is the sum of two squares in $F$.

**(a)** Let $f_1(x)$ and $f_2(x)$ be irreducible polynomials over a finite field $\F_p$, where $p$ is a prime number. Suppose that $f_1(x)$ and $f_2(x)$ have the same degrees. Then show that fields $\F_p[x]/(f_1(x))$ and $\F_p[x]/(f_2(x))$ are isomorphic.

**(b)** Show that the polynomials $x^3-x+1$ and $x^3-x-1$ are both irreducible polynomials over the finite field $\F_3$.

**(c)** Exhibit an explicit isomorphism between the splitting fields of $x^3-x+1$ and $x^3-x-1$ over $\F_3$.