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?
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Strictly speaking, because the problem said "irreduceable over the integers", it is because there are no integers satisfying [itex]x^2+1= 0[/itex]. 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 [itex]x^2- 2[/itex] is irreduceable over the integers.