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  1. How do I show that a polynomial is irreducibel?
  2. How do I show that $x^2+1$ is irreducible over the field $F_p$ where $p \equiv 3 \mod 4$?

My guess for number 1) is that inserting all numbers $x$ from zero to $p-1$ into the formula, the polynomial is irreducible as long as $f(x)$ is never zero. That somehow only works for irreducible polynomials of degree 2 or 3.

No idea how to solve exercise 2)

Korona Genius
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  • @RichardD.James thank you, I corrected it! – Korona Genius May 25 '20 at 06:42
  • The congruence $ x^{2}\equiv -1{\bmod {p}} $ is solvable if and only if $ p\equiv1\bmod4$ – J. W. Tanner May 25 '20 at 06:45
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    The second question likely has several answers already on MSE: see https://math.stackexchange.com/questions/2389963/argument-for-why-a2-1-is-never-divisible-by-a-3-mod-4-integer?rq=1 – Aravind May 25 '20 at 06:56
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    The second answer at https://math.stackexchange.com/questions/215469/characterization-of-irreducible-polynomials-over-finite-fields?rq=1 answers the first question. – Aravind May 25 '20 at 06:59