Difference between revisions of "Schonemann's criterion"

Revision as of 11:43, 18 December 2009

For a polynomial q denote by $q^*$ the residue of $q$ modulo $p$ . Suppose the following conditions hold:

Then $k$ is irreducible in $\mathbb{Q}[X]$ .

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 For a polynomial q denote by <math>q^*</math> the residue of <math>q</math> modulo <math>p</math>.
 Suppose the following conditions hold:
-<list>
+* <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>.
-<it> <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>
-<it> <math>\text{deg}(f^n)>\text{deg}(g)</math>
+* <math>k</math> is primitive
-<it> <math>k</math> is primitive
+* <math>f^*</math> is irreducible in <math>\mathbb{F}_p[X]</math>.
-<it> <math>f^*</math> is irreducible in <math>\mathbb{F}_p[X]</math>.
+* <math>f^*</math> does not divide <math>g^*</math>.
-<it> <math>f^*</math> does not divide <math>g^*</math>.
-<\list>
 Then <math>k</math> is irreducible in <math>\mathbb{Q}[X]</math>.
 See also [[Eisenstein's criterion]].