Schonemann's criterion
For a polynomial 
denote by 
the residue of modulo 
.
Suppose the following conditions hold:
with 
, 
prime, and ![$f,g\in \mathbb{Z}[X]$](//latex.artofproblemsolving.com/7/b/8/7b8b5aecb8346c6528ad1b520a84656bb2e49a05.png)
.
.
is primitive.
is irreducible in ![$\mathbb{F}_p[X]$](//latex.artofproblemsolving.com/1/b/6/1b69862785353c9500fcec83ebfbe95348950d6c.png)
.- does not divide
.
with 
, ![$f,g\in \mathbb{Z}[X]$](http://latex.artofproblemsolving.com/7/b/8/7b8b5aecb8346c6528ad1b520a84656bb2e49a05.png)
.
.
is irreducible in ![$\mathbb{F}_p[X]$](http://latex.artofproblemsolving.com/1/b/6/1b69862785353c9500fcec83ebfbe95348950d6c.png)
.
.Then 
is irreducible in ![$\mathbb{Q}[X]$](http://latex.artofproblemsolving.com/f/3/d/f3dc4b61f66a97c235268da5643f5bebbc69c9bd.png)
.
See also Eisenstein's criterion.
