I want to construct an irreducible polynomial of degree 2 over Fp where p is a prime that can be written as 4k+1. My attempt is as follow: we can assume that this polynomial is of the form x2+ax+b for some a,b∈Fp. So for all λ∈Fp, p doesn't divide λ2+aλ+b. It follows that λ2 is not equal to aλ+bmod. If we can find some a,b \in {\mathbb{F}_p} such that a\lambda + b is a nonresidue for all \lambda \in {F_p}, it is ok. But I cannot. I wait your response.
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