What are the irreducible polynomials in Z5[x]?

[missavvy]

Homework Statement

Factor f(x) = 3x⁴ + 2 into a product of irreducible polynomials in Z₅[x]

Homework Equations

The Attempt at a Solution

I don't get it. I tried dividing it using the division logarithm, but then I can only get it to a point where it's like, 3(x-1)(..) <- polynomial of degree 3
Just simply plugging in values of Z₅ doesn't seem to work..

I know there's some sort of trick to use.. I don't really understand how to factor f(x) using the different degrees. :S My textbook does a poor job of explaining it, without any concrete examples for me to go by, and I tried googling it but only saw an unanswered question.

If anyone can explain or direct me to some websites that explain how to do this, that would be great.
:)

 

[lanedance]

for clarity, how about constructing a table something like
x _ x^4 _ 3x^4 _ 3x^4 mod 5 _ (3x^4 +2)mod5
0 ...
1 ...
2 ...
..

 

[missavvy]

and fill in all the values? for each element in Z5?

 

[lanedance]

well just to get a handle on how it all works in Z5, this latex formatting may help
$$\begin{matrix}<br /> x & x^4 & 3x^4 & 3x^4 +2 & (3x^4 +2)mod5\\<br /> 0 & 0 & 0 & 2&2 \\<br /> 1& 1& 3&5 & 0 \\<br /> ...& & & & & \\<br /> \end{matrix}$$

 

[lanedance]

this may also help
http://en.wikipedia.org/wiki/Irreducible_polynomial#Simple_examples
http://en.wikipedia.org/wiki/Finite_field_arithmetic#Addition_and_subtraction

 

[missavvy]

Got it thanks!