Primitivity Rules

Number Theory Level 4

For any positive integer n n , we define the cyclotomic polynomial Φ n ( x ) = ∏ ( x − w ) \Phi_n(x)=\prod(x-w) , where the product is taken over all primitive nth roots of unity, w w . What can we say about the coefficients of the cyclotomic polynomials?

Extra Credit Question: Does the number 2 appear as a coefficient of a cyclotomic polynomial?

All the coefficients are integers All the coefficients are real, and they are irrational in some cases The coefficients are (non-real) complex in some cases All the coefficients are rational, and they fail to be integers in some cases

Otto Bretscher by Otto Bretscher , 65, USA

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