# Cyclotomic Polynomials

### Jack Shotton

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.

#### Sources:

You can start by looking at the wikipedia page! Most algebra textbooks will have something on cyclotomic polynomials (e.g. Lang's Algebra, Dummit and Foote) and that is where we will start. My hope is that (with guidance) you will find your own references.

email: Jack Shotton