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x^n=(x^2+1)^n if and only if ...

Source: Romanian National Olympiad 2024 - Grade 12 - Problem 4

April 3, 2024
finite fieldsabstract algebra

Problem Statement

Let L\mathbb{L} be a finite field with qq elements. Prove that:
a) If q≡3(mod4)q \equiv 3 \pmod 4 and n≥2n \ge 2 is a positive integer divisible by q−1,q-1, then xn=(x2+1)nx^n=(x^2+1)^n for all x∈L×.x \in \mathbb{L}^{\times}.
b) If there exists a positive integer n≥2n \ge 2 such that xn=(x2+1)nx^n=(x^2+1)^n for all x∈L×,x \in \mathbb{L}^{\times}, then q≡3(mod4)q \equiv 3 \pmod 4 and q−1q-1 divides n.n.
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