Difference between revisions of "Schonemann's criterion"
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Suppose the following conditions hold: | Suppose the following conditions hold: | ||
* <math>k=f^n+pg</math> with <math>n\geq 1</math>, <math>p</math> prime, and <math>f,g\in \mathbb{Z}[X]</math>. | * <math>k=f^n+pg</math> with <math>n\geq 1</math>, <math>p</math> prime, and <math>f,g\in \mathbb{Z}[X]</math>. | ||
| − | * <math>\text{deg}(f^n)>\text{deg}(g)</math> | + | * <math>\text{deg}(f^n)>\text{deg}(g)</math>. |
| − | * <math>k</math> is primitive | + | * <math>k</math> is primitive. |
* <math>f^*</math> is irreducible in <math>\mathbb{F}_p[X]</math>. | * <math>f^*</math> is irreducible in <math>\mathbb{F}_p[X]</math>. | ||
* <math>f^*</math> does not divide <math>g^*</math>. | * <math>f^*</math> does not divide <math>g^*</math>. | ||
Revision as of 10:44, 18 December 2009
For a polynomial q 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.
