Question

Prove that every element in a finite field can be written as the sum of two squares.

Short answer

Every element in a finite field can be written as the sum of two squares.

Step-by-step solution

Definition of finite field

A finite field is a field that contain a finite number of elements.

Step-2: Showing that every element in a finite field can be written as the sum of two squares

Let F be a finite field and its characteristic is p and $\left|F\right|=p^n$. Define by $\mathrm{\varphi }:\mathrm{F}\to \mathrm{F}$as $\mathrm{\varphi }\left(\mathrm{x}\right)={\mathrm{x}}^2,\mathrm{p}=2$.

Then $\mathrm{\varphi }$is isomorphic and so for any $u\in F$there is a $v\in F$with $u=v^2+0^2$.

If p > 2 then for all $x,y\in F$as $x^2=y^2$.

This implies that:

$$\begin{gathered} \left(x+y\right)\left(x-y\right)=0 \\ y=x,y=-x \end{gathered}$$

And so :

$\left|ln\mathrm{\varphi }\right|\ge \frac{p^2+1}{2}$

Now let $m=\frac{p^n+1}{2}$and choose distinct in $x_1^2,x_2^2..........{\mathrm{x}}_{\mathrm{n}}^{\mathrm{n}}\mathrm{in}F$.

Since $2m>p^n$ there exist j and k such that:

$$\begin{gathered} x_j^2=u-x_k^2 \\ u=x_j^2+x_k^2 \end{gathered}$$

From this the required condition is satisfy