1. ## Irreducible Polynomial

Hi guys, im stuck on a question which is:

Prove that f(x)= x^4 + x + 2 is irreducible over Z3

Thanks.
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Note: I show that the linear factors for f(a) over Z3 are not equal to zero,
But how would i determine quadratic factors?(help solve) thanks.

2. Hello,
Originally Posted by furnis1
Hi guys, im stuck on a question which is:

Prove that f(x)= x^4 + x + 2 is irreducible over Z3

Thanks.
----
Note: I show that the linear factors for f(a) over Z3 are not equal to zero,
But how would i determine quadratic factors?(help solve) thanks.
In $\mathbb{Z}_3$, $x^3=x$, by Fermat's little theorem.

Hence $f(x)=x^2+x+2$

Then you can just successively let $x=[0]$, $x=[1]$ and $x=[2]$, and see if $f(x)=0$ in any case.

3. Originally Posted by Moo
Hello,

In $\mathbb{Z}_3$, $x^3=x$, by Fermat's little theorem.

Hence $f(x)=x^2+x+2$

Then you can just successively let $x=[0]$, $x=[1]$ and $x=[2]$, and see if $f(x)=0$ in any case.
First, $x^2+x+2\not = x^4+x+2$ !
Second, just because a polynomial has no zeros does not make it irreducible.

4. Originally Posted by ThePerfectHacker
First, $x^2+x+2\not = x^4+x+2$ !
Second, just because a polynomial has no zeros does not make it irreducible.
But if $\mathbb{Z}_3$ is indeed $\mathbb{Z}/3\mathbb{Z}$, then $x^4=x^2$, isn't it ?

And for a polynomial of degree 2, if there is no zero, then it's irreducible (if we're not talking about complex numbers of course !)

5. Originally Posted by Moo
But if $\mathbb{Z}_3$ is indeed $\mathbb{Z}/3\mathbb{Z}$, then $x^4=x^2$, isn't it ?
No!

Let $f(x) = x^4 + x + 2$ and $g(x) = x^2 + x + 2$.
Then it turns out that $f(\alpha) = g(\alpha)$ for each $\alpha \in \mathbb{Z}_3$.
However, the two polynomials are not the same!

What is a polynomial? Here is one formal definition. Let $F$ be a field. Define $\widehat F = F\times F\times F\times F \times ...$ - an infinite tuple of coordinates. A polynomial $f$ is $f\in \widehat F$ such that $f_i$ (at the $i$-th coordinate) satisfies $f_i = 0$ for all but finitely many $i$. So when we write $x+x^2$ we mean $(0,1,1,0,0,0,0,...)$ and when we write $1+2x^2$ we mean $(1,0,2,0,0,0,0,...)$ and so on. We say two polynomial are equal if and only if $f=g$ i.e. if and only if the two coordinates match.

I think your confusion, dear, is that you are confusing polynomial for polynomial function. A polynomial is an abstract meaning as defined above. A polynomial function is a function consisting of powers of x. Now if two polynomial functions agree at all points then the two polynomial functions are equal. However, if two polynomials (abstract) agree at all the points does not mean they are the same.

I give you an example. The polynomial, $x^p - x - 1$ (known as Artin-Schreir polynomial - kinda famous) over $\mathbb{Z}_p$ can be shown to be irreducible, however, $f(\alpha) = -1$ for all $\alpha\in \mathbb{Z}_p$. This does not mean that $x^p - x - 1 = -1$.