Final answer:
To prove that every element in a finite field can be expressed as the sum of two squares, we can use the fact that every non-zero element in a finite field has a multiplicative inverse. By finding two numbers, a and b, we can express x as the sum of their squares.
Step-by-step explanation:
To prove that in every finite field F, every element of F is the sum of two squares, we can use the fact that in a finite field, every non-zero element has a multiplicative inverse. Let x be an element of F. We can express x as the sum of two squares by finding two numbers a and b such that a^2 + b^2 = x.
Since every non-zero element in F has a multiplicative inverse, we can find y such that y^2 = 1.
Now, we can express x as the sum of two squares: x = (ay)^2 + (by)^2 = a^2y^2 + b^2y^2 = (a^2 + b^2)y^2 = (a^2 + b^2)(1) = a^2 + b^2.