How to factor polynomials in $\mathbb{Z}_n$?

Question

How to factor a certain polynomial over $Zn$.

for example factor the following polynomial into irreducible polynomials in $Z5$:

$X^3+X^2+X-1$

or factor the following polynomial into irreducible polynomials in $Z2$:

$X^4+X+1$

is there a certain method (algorithm) I can follow?

Please help I'm stuck and i really need the help!

thank you in advance!!

See http://en.wikipedia.org/wiki/Factorization_of_polynomials_over_finite_fields. — Martín-Blas Pérez Pinilla, Dec 18 '14 at 11:45
The proper term is over rather than in the ring of scalars. May we assume that $n$ is a prime number, or is that not assumed? — Marc van Leeuwen, Dec 18 '14 at 13:05

Mizi · Answer 1 · 2014-12-18T13:18:00.623

2

let $ f $ be a irreducible polynomial over finite field $\Bbb{F}_q$ and $ \alpha$ is a zero of $f$. let $ d=\mathrm{deg}(f)$. then degree of $\Bbb{F}_q(\alpha)$ is $d$ and the zero is also zero of $ x^{q^d}-x$. therefore all irreducible polynomial with degree $d$ is factor of $x^{q^d}-x$. If $f$ is not a factor of $x^{q^d}-x$, then $f$ is reducible.

edited Dec 18 '14 at 13:18

answered Dec 18 '14 at 13:06

Mizi

186
5

Timbuc · Answer 2 · 2014-12-18T11:58:09.257

0

The first one is easily proven to be irreducible since a polynomial of degree $\;\le 3\;$ is reducible over some field iff it has a root in that field.

For the second one observe that

$$x^4+x^2+1=(x^2+x+1)^2\in\Bbb F_2[x]$$

edited Dec 18 '14 at 11:58

answered Dec 18 '14 at 11:43

Timbuc

34,795

But how would I do the factorization using a methodology not just by observation? Thank you for the answer! – ZZZZZZzzzzzz Dec 18 '14 at 12:19
1

There are several algorithms, e.g., the Berlekamp algorithm. – Dietrich Burde Dec 18 '14 at 12:51

How to factor polynomials in $\mathbb{Z}_n$?

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