Difference between revisions of "Schonemann's criterion"
Silversheep (talk | contribs) |
m (latex \deg) |
||
(4 intermediate revisions by 4 users not shown) | |||
Line 1: | Line 1: | ||
− | + | If <math>f(x)\in \mathbb{Z}[x]</math> | |
− | + | * <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) \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. | |
− | + | ||
− | + | ==Proof== | |
− | + | We know that <math>f(x)</math> is monic, so <math>\deg f=n\deg g</math> and we may assume that <math>g(x)</math> is monic. Assume <math>f(x)=p(x)q(x)</math>, where <math>p(x), q(x)\in \mathbb{Z}[x]</math>. Since <math>f(x)=p(x)q(x) \pmod{p}</math>, we get <math>\overline{F}=f(x) \pmod{p}</math>, so <math>g(x)^n \pmod{p}=\overline{P}\cdot \overline{Q}</math>. Therefore, we have <math>p(x)=g(x)^r+pP_1(x)</math> and <math>q(x)=g(x)^{n-r}+pQ_1(x)</math> for some <math>P_1(x)</math> and <math>Q_1(x)</math>. Therefore, | |
+ | <cmath>g(x)^n+ph(x)=f(x)=p(x)q(x)=g(x)^n+p(P_1(x)g(x)^{n-r}+Q_1(x)g(x)^r+pP_1(x)Q_1(x))</cmath> | ||
+ | This means that <math>h(x)=P_1(x)g(x)^{n-r}+Q_1(x)g(x)^r+pP_1(x)Q_1(x)</math>, which means that <math>g(x)\pmod{p}\vert h(x)\pmod{p}</math>, a contradiction. This means that <math>f(x)</math> is irreducible. | ||
+ | |||
+ | {{stub}} | ||
See also [[Eisenstein's criterion]]. | See also [[Eisenstein's criterion]]. |
Latest revision as of 17:03, 31 October 2023
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
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.
This article is a stub. Help us out by expanding it.
See also Eisenstein's criterion.