ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 10 Oct 2017 03:49:57 +0200Another way to determine if a polynomial is irreducible over a ring, is to check if the zeros of the polynomial are in the ring, ie, does p have a linear factor.https://ask.sagemath.org/question/39079/another-way-to-determine-if-a-polynomial-is-irreducible-over-a-ring-is-to-check-if-the-zeros-of-the-polynomial-are-in-the-ring-ie-does-p-have-a-linear/ Another way to determine if a polynomial is irreducible over a ring, is to check if the zeros of the polynomial are in the ring, ie, does p have a linear factor.
# For Z_5, write a "for" loop to check if p evaluated at every element of Z_5 is zero.
Mon, 09 Oct 2017 06:11:00 +0200https://ask.sagemath.org/question/39079/another-way-to-determine-if-a-polynomial-is-irreducible-over-a-ring-is-to-check-if-the-zeros-of-the-polynomial-are-in-the-ring-ie-does-p-have-a-linear/Comment by dan_fulea for <p>Another way to determine if a polynomial is irreducible over a ring, is to check if the zeros of the polynomial are in the ring, ie, does p have a linear factor.</p>
<h1>For Z_5, write a "for" loop to check if p evaluated at every element of Z_5 is zero.</h1>
https://ask.sagemath.org/question/39079/another-way-to-determine-if-a-polynomial-is-irreducible-over-a-ring-is-to-check-if-the-zeros-of-the-polynomial-are-in-the-ring-ie-does-p-have-a-linear/?comment=39094#post-id-39094"The zeros" are all the zeros?
A linear factor is only one factor?
If a (monic) polynomial over a (commutative) ring (with one) has a linear factor, than it is of course reducible. But the converse is obviously false, e.g. $(x^2+2)^2$ over integers, even over reals, or as is over some finite field...
This looks like homework, please insert the own trials and errors.Tue, 10 Oct 2017 03:49:57 +0200https://ask.sagemath.org/question/39079/another-way-to-determine-if-a-polynomial-is-irreducible-over-a-ring-is-to-check-if-the-zeros-of-the-polynomial-are-in-the-ring-ie-does-p-have-a-linear/?comment=39094#post-id-39094