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Finite Field with $q \ne 1 (mod 4)$ Elements

Source: Romanian District Olympiad 2018 - Grade XII - Problem 4

March 10, 2018

finite fieldssuperior algebra

Problem Statement

Let

n

and

q

be two natural numbers,

n\ge 2

q\ge 2

and

q\not\equiv 1 (\text{mod}\ 4)

and let

K

be a finite field which has exactly

q

elements. Show that for any element

a

from

K

, there exist

x

and

y

K

such that

a = x^{2^n} + y^{2^n}

. (Every finite field is commutative).