Cyclotomic Polynomials

For any natural number \(n\), the primitive \(n\)th roots of unity are \(e^{2 \pi i k/n}\) for all \(k\) such that \(1 \le k \le n\) and \(\mathrm{gcd}(k,n) = 1\). The \(n\)th cyclotomic polynomial \(\Phi_n\) is then \[\Phi_n(x) = \prod_{\substack{1 \le k \le n \\ \mathrm{gcd}(k,n) = 1}}(x - e^{2 \pi i k/n}).\] It has integer coefficients and is irreducible --- not obvious, particularly the second! There is a factorization \[x^n - 1 = \prod_{d \mid n} \Phi_d(x).\] This can also be taken as a (recursive) definition of \(\Phi_n(x)\).

Here are the first few: \[\begin{align} \Phi_1(x) & = x - 1\\ \Phi_2(x) & = x + 1 \\ \Phi_3(x) & = x^2 + x + 1 \\ \Phi_4(x) & = x^2 + 1 \\ \Phi_5(x) &= x^4 + x^3 + x^2 + x + 1 \\ \Phi_6(x) &= x^2 - x + 1 \\ \Phi_7(x) &= x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 \\ \Phi_8(x) &= x^4 + 1 \\ \Phi_9(x) &= x^6 + x^3 +1\\ & \vdots \\ \Phi_{15}(x) &= x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 \end{align}\]

Spot any patterns yet? Are the coefficients always \(\pm 1\)? What is \(\Phi_p\) for \(p\) a prime (maybe you did this in Algebra II)? What about powers of primes? Products of two primes? Products of three primes (for example, 105)? What is \(\Phi_n(1)\)? How could you compute \(\Phi_n\) efficiently? Hopefully you will be able to come up with some of your own questions, and we will try to answer them!

Prerequisites:

Algebra II and Elementary Number Theory II.

Corequisites:

None. Would go well with Number Theory III or Galois Theory III.

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