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