## Irreductible polynomial with paired roots

Suppose is an irreductible polynomial which has a complex root such that is also a root for . Prove that for any other root of , is also a root for .

**Proof: **We may assume WLOG that the leading coefficient of is . Then is the minimal polynomial of and . Of course , since is irreductible. Moreover, since is the minimal polynomial for we get that . Because is not a root for we must have , which means that for any root of , we have that is also a root of .

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Categories: Algebra, Olympiad, Problem Solving
irreductibility, irreductible, polynomial

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