Not as complex as it looks

Number Theory Level 4

For any positive integer $n$ , we define the cyclotomic polynomial $\Phi_n(x)=\prod(x-w)$ , where the product is taken over all primitive $n$ th roots of unity, $w$ .

Which of the statements (a), (b), (c), (d), (e), in this order, is the FIRST to be true?

All coefficients of all cyclotomic polynomials are

(a) 0, 1, or -1

(b) integers

(d) real

(e) complex

c a d e b

1 solution

Otto Bretscher
Oct 15, 2015

Since every $n$ th root of unity is a primitive root of unity for exactly one divisor $d$ of $n$ , we can write $\Phi_n(x)=\frac{x^n-1}{\prod_{d|n,d<n}\Phi_d(x)}$ By strong induction on $n$ , we can assume that the denominator is a monic polynomial with integer coefficients. Based on the algorithm of long division, we can conclude that $\Phi_n(x)$ has integer coefficients as well.

For small $n$ , the coefficients of $\Phi_n(x)$ are all 0, 1, or -1, but this is not the case in general. For example, $\Phi_{105}$ has two coefficients -2

Not as complex as it looks

1 solution

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