Proving Irreducibility of Polynomials over the Integers

[RTH001]

If i have to show a polynomial x^2+1 is irreduceable over the integers, is it enough to show that X^2 + 1 can only be factored into (x-i)(x+i), therefore has no roots in the integers, and is subsequently irreduceable?

 

[Dick]

I wouldn't drag imaginary numbers into this. If x^2+1 is reducible over the integers then it would split into two linear factors with integer coefficients. Can you argue why that can't happen?

 

[RTH001]

is it because x^2+1 has no real solutions?

 

[HallsofIvy]

Strictly speaking, because the problem said "irreduceable over the integers", it is because there are no integers satisfying $x^2+1= 0$. Of course, since the integers are a subset of the real numbers yours is a sufficient answer. But your teacher might call your attention to the difference by (on a test, perhaps) asking you to show that $x^2- 2$ is irreduceable over the integers.