How to know if this polynomial is irreducible

Asked Jan 19 '23 at 15:21

Active Jan 19 '23 at 15:29

Viewed 74 times

I've been working on this problem and I'm stuck on this part, I need to factorize this polynomial in the field of integers modulo 5

$f(x)=(x^5-x)^2-1$

And so far I've gotten that by difference of squares that it can be factored into $$f(x)=(x^5-x-1)(x^5-x+1)$$

However from here I cannot really tell for sure this two polynomials are irreducible, I know (Because of the solution that they are), but I can't justify it, I've thought of trying to use Eisenstein but it's not applicable and there are too many irreducible polynomials to try them out one by one

Is there something I'm missing?

Thank you in advance!

edited Jan 19 '23 at 15:29

asked Jan 19 '23 at 15:21

Ramiro genta

2

A reducible polynomial of degree 5 must have a linear or quadratic factor. Check both these options. – Paramanand Singh Jan 19 '23 at 15:27
I'm sorry but why only those two factors? Couldn't I also reduce it in one polynomial of degree 3 and other of degree 2? – Ramiro genta Jan 19 '23 at 15:28
2

Degree 2 factor is what I call a quadratic factor. – Paramanand Singh Jan 19 '23 at 15:39
So what I should do is try polynomials and see that i can't divide them? – Ramiro genta Jan 19 '23 at 15:49
2

Clearly there are no linear factors . There are, I think, $10$ irreducible quadratics mod $5$. Not far too many to try, but tedious. Perhaps easier would be to crunch through the euclidean algorithm and see whether $hcf(x^5 - x -1, x^{25}-x)$ is non-trivial, not more than half an hour's work. – ancient mathematician Jan 19 '23 at 15:53
1

See related https://math.stackexchange.com/q/81583/72031 and let me know if your question can be closed as a duplicate of this. – Paramanand Singh Jan 20 '23 at 07:47

How to know if this polynomial is irreducible

0 Answers0