Category:Examples of Use of Schönemann-Eisenstein Theorem

This category contains examples of use of Schönemann-Eisenstein Theorem.

Let $\map f x = a_d x^d + a_{d - 1} x^{d - 1} + \dotsb + a_0 \in \Z \sqbrk x$ be a polynomial over the ring of integers $\Z$.

Let $p$ be a prime such that:

$(1): \quad p \divides a_i \iff i \ne d$

$(2): \quad p^2 \nmid a_0$

where $p \divides a_i$ signifies that $p$ is a divisor of $a_i$.

Then $f$ is irreducible in $\Q \sqbrk x$.