Thread: irreducible polynomial

I'm just now trying to do another related exam question:

Show that g = X^3 + 2X - 1 is irreducible over rationals.

If x = b/c, b must be a factor of -1, c must be a factor of 1.

Possible factors of -1: +1, -1
Possible factors of 1: +1, -1

List of possible rational roots:

Direct checking shows that none of these are a root and hence g has no rational root.

Therefore since there are no rational roots g cannot be irreducible over the rationals.


Is this the right approach for this question?

Thanks in advance!

Originally Posted by hunkydory19

I'm just now trying to do another related exam question:

Show that g = X^3 + 2X - 1 is irreducible over rationals.

If x = b/c, b must be a factor of -1, c must be a factor of 1.

Possible factors of -1: +1, -1
Possible factors of 1: +1, -1

List of possible rational roots:

Direct checking shows that none of these are a root and hence g has no rational root.


Therefore since there are no rational roots g cannot be irreducible over the rationals.


Is this the right approach for this question?

Thanks in advance!

This seems like the right approach to me. In fact, I don't know how else you would do it. Your conclusion, however, should be that g is irreducible. I'm sure you just made a typo.

Originally Posted by hunkydory19

I'm just now trying to do another related exam question:

Show that g = X^3 + 2X - 1 is irreducible over rationals.

If x = b/c, b must be a factor of -1, c must be a factor of 1.

Possible factors of -1: +1, -1
Possible factors of 1: +1, -1

List of possible rational roots:

Direct checking shows that none of these are a root and hence g has no rational root.

Therefore since there are no rational roots g cannot be irreducible over the rationals.


Is this the right approach for this question?

Thanks in advance!

Yes. You need to remember a polynomial is reducible over Q if and only if it is reducible over Z. This is why you are allowed to restrict your attention to
integers.

Well someone else told me that I should be using Gauss's lemma, so I wasn't sure if it was OK to use this method, but since it is I'll definitely be using this as it's much easier!

Thank you for your help!