Home > Algebra, Olympiad, Problem Solving > Irreductible polynomial with paired roots

Irreductible polynomial with paired roots


Suppose f \in \Bbb{Q}[X] is an irreductible polynomial which has a complex root a such that -a is also a root for f. Prove that for any other root b of f, -b is also a root for f.

Proof: We may assume WLOG that the leading coefficient of f is 1. Then f is the minimal polynomial of a and -a. Of course a \neq 0, since f is irreductible. Moreover, since \pm f(-x) is the minimal polynomial for -a we get that \pm f(-x)=f(x). Because 0 is not a root for f we must have f(x)=f(-x), which means that for any root b of f, we have that -b is also a root of f.

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