Exercise 1842Number theory › PrimesM1Added 3 Nov 2026

Primes and polynomial irreducibility

#irreducibility#primes#polynomial

Statement

Show that if $p$ is prime, then $X^2-2$ is irreducible over $\mathbb{F}_p$ iff $2$ is a quadratic non-residue mod $p$ .

Hint

A quadratic is reducible iff it has a root, i.e. iff $2$ is a square mod $p$ .

Show solutionHide solution

Root condition. $X^2-2$ has a root in $\mathbb{F}_p$ iff $2$ is a QR mod $p$ .

Irreducibility. A quadratic over a field is irreducible iff it has no root. So $X^2-2$ is irreducible iff $2$ is a QNR mod $p$ .

Examples. $p=3$ : $2$ is QNR, irreducible. $p=7$ : $2$ is QR ( $3^2=9\equiv2$ ), reducible: $X^2-2=(X-3)(X+3)\pmod7$ .

Conclusion. The irreducibility is governed by the Legendre symbol $\left(\frac{2}{p}\right)$ .