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Irreducible function on Q[z]

Let be irreducible with degree . If has a root on the unit circle then is even and

Solution

Let be a root of with . Since has real coefficients is also a root of . The product is a polynomial in of degree ( its leading coefficient is ) with root . By the irreducibility of we have

(1)

for some non zero rational number . Setting we have that . Since   , by our hypotheses , hence . Setting we get that and because we deduce that is even.

Note: The above tells us that can be expressed in terms of .