Prove the polynomial f(x)=x^2-q is irreducible in F_p[x]?

[MustardTiger]

Prove the polynomial

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[daon2]

If p and q are prime numbers such that p is not a quadratic residue mod q. Show that if pq=-1 mod 4 then the polynomial f(x)=x^2-q is irreducible in F_p[x].

A quadratic is irreducible in F_p if and only if it has no roots in F_p (this is also true for cubic polynomials, since, if it factors nontrivially, it factors into a quadratic and a linear polynomial).

There is a relatively straight-forward way to show that if your polynomial factors than q=-1 (mod p), and x^2+1 factors over F_p[x] if and only if p=-1(mod 4).