Difference between revisions of "Schonemann's criterion"
m (Fixed several typos where f's needed to be g's.) |
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* <math>f(x)</math> is monic | * <math>f(x)</math> is monic | ||
* <math>g(x), h(x)\in \mathbb{Z}[x]</math>, a prime <math>p</math> and an integer <math>n</math> such that <math>f(x)=g(x)^n+ph(x)</math> | * <math>g(x), h(x)\in \mathbb{Z}[x]</math>, a prime <math>p</math> and an integer <math>n</math> such that <math>f(x)=g(x)^n+ph(x)</math> | ||
− | * <math>g(x) \pmod{p}</math> is | + | * <math>g(x) \pmod{p}</math> is a irreducible polynomial in <math>\mathbb{F}_p</math> and does not divide <math>h(x) \pmod{p}</math> |
then <math>f(x)</math> is irreducible. | then <math>f(x)</math> is irreducible. | ||
Revision as of 12:25, 14 June 2021
If
- is monic
- , a prime and an integer such that
- is a irreducible polynomial in and does not divide
then is irreducible.
Proof
We know that is monic, so deg deg and we may assume that is monic. Assume , where . Since , we get , so . Therefore, we have and for some and . Therefore, This means that , which means that , a contradiction. This means that is irreducible.
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See also Eisenstein's criterion.