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!