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# Irreducible factors of a polynomial

Let and let

Find all irreducible factors of .

Solution

Setting we note that

Hence

(1)

It’s clear that is irreducible over . Now, for let be the -th cyclotomic polynomial. Using well-known properties of , we have

Thus is irreducible over because cyclotomic polynomials are irreducible over . Hence, by ( 1 ) has exactly irreducible factors and they are .

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