Cyclotomic polynomial for any positive integer

Question

i have searched about it but i couldn't get a clear simpler definition of this polynomial, so i need to understand more about it because i am asked how to construct it? And is this polynomial irreducible or reducible over Q? also i wanted to know some examples for this polynomial if n=1,2,3,....20 (maybe Theorems and their proofs relative to this subject could help me to understand more). i'd be really glad

Answer 1

The cyclotomic polynomial of order $n$ is the polynomial whose complex roots are all primitive $n$-th roots of unity, so $n$-th roots of unity which aren't $k$-th roots of unity for any $0 < k < n$.

Now to get the primitive $n$-th roots of unity, the idea is usually to start with all $n$-th roots of unity, then remove all nonprimitive ones, right? Translating that into roots of polynomials, to get the cyclotomic polynomial $\Phi_n$, you start with $X^n - 1$, then divide by $\Phi_k$ for all $k \mid n$ with $k < n$, thereby eliminating all nonprimitive roots. Cyclotomic polynomials are indeed irreducible over $\mathbb Q$.

Consider the example $n = 6$. You have $\Phi_1 = X-1$, $\Phi_2 = X + 1$, $\Phi_3 = X^2+X+1$. So $$\Phi_6 = (X^6 - 1)/(\Phi_1\Phi_2\Phi_3) = X^2-X+1.$$

The Wikipedia article on cyclotomic polynomials has a list of cyclotomic polynomials for small $n$.