What are the irreducible polynomials in Z5[x]?

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In summary, the conversation is about factoring the polynomial 3x4 + 2 into irreducible polynomials in Z5[x]. The individual has attempted to divide using the division logarithm, but is having trouble understanding how to factor using different degrees. They have asked for clarification and direction to resources, and a helpful response suggests constructing a table and provides links to examples and information on finite field arithmetic. Eventually, the individual states that they have understand the concept and thanks for the help.
  • #1
missavvy
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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.
:)
 
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  • #2
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 ...
..
 
Last edited:
  • #3
and fill in all the values? for each element in Z5?
 
  • #4
well just to get a handle on how it all works in Z5, this latex formatting may help
[tex]
\begin{matrix}
x & x^4 & 3x^4 & 3x^4 +2 & (3x^4 +2)mod5\\
0 & 0 & 0 & 2&2 \\
1& 1& 3&5 & 0 \\
...& & & & & \\
\end{matrix}
[/tex]
 
  • #6
Got it thanks!
 
Last edited:

1. What is an irreducible polynomial in Zn?

An irreducible polynomial in Zn is a polynomial that cannot be factored into smaller polynomials over the finite field Zn. In other words, it is a polynomial that cannot be written as the product of two or more non-constant polynomials.

2. How do you determine if a polynomial is irreducible in Zn?

To determine if a polynomial is irreducible in Zn, you can use the Eisenstein's criterion. This states that if a polynomial has a prime number that divides all its coefficients except the leading coefficient, and the square of that prime does not divide the constant term, then the polynomial is irreducible.

3. Why is the concept of irreducible polynomials important in Zn?

Irreducible polynomials in Zn are important because they form the building blocks for creating finite fields. They are also used in various areas of mathematics, such as coding theory and cryptography.

4. Can an irreducible polynomial in Zn have multiple roots?

No, an irreducible polynomial in Zn cannot have multiple roots. This is because in a finite field, every polynomial of degree n has at most n roots. If an irreducible polynomial had multiple roots, it would have more than n roots, which is not possible in a finite field.

5. How are irreducible polynomials in Zn used in coding theory?

In coding theory, irreducible polynomials in Zn are used to construct error-correcting codes. These codes are used to detect and correct errors that may occur during data transmission. The properties of irreducible polynomials make them ideal for constructing these codes, as they have a low probability of producing incorrect results.

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