The cyclotomic polynomials ${\mathrm{\Phi }}_n\in \mathbb{Z}[t]$, defined by ${\prod }_{d\mid n}{\mathrm{\Phi }}_d=t^n-1$

The cyclotomic polynomials ${\mathrm{\Phi }}_n\in \mathbb{Z}[t]$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution) are defined by recursion on $n\ge 1$:

${\mathrm{\Phi }}_1:=t-1,{\mathrm{\Phi }}_n:=\frac{t^n-1}{{\displaystyle \prod _{\begin{array}{c}d\mid n\\ 0<d<n\end{array}}{\mathrm{\Phi }}_d}}(n\ge 2),$

the divisors being the positive divisors of $n$ (Divisibility in $\mathbb{Z}$: $d\mid a$ when $a=dq$ for some integer $q$). The product is formed in the commutative ring $\mathbb{Z}[t]$ (Polynomial convolution makes $R[x]$ a commutative ring containing $R$ as its constant subring): multiply the finitely many factors in any enumeration of the divisor set. Associativity and commutativity make the result independent of that enumeration.

What the fraction means. The denominator $P_n:={\prod }_{d\mid n,d<n}{\mathrm{\Phi }}_d$ is a product of monic polynomials in $\mathbb{Z}[t]$, hence itself monic (Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree), so Division by a monic polynomial over a commutative ring supplies a unique pair $q,r\in \mathbb{Z}[t]$ with $t^n-1=q{P}_n+r$ and $r=0$ or $\mathrm{deg}r<\mathrm{deg}{P}_n$. The definition sets ${\mathrm{\Phi }}_n:=q$, and asserts $r=0$. That assertion, together with the consequences that each ${\mathrm{\Phi }}_n$ is monic of degree $\phi (n)$ and that

${\prod }_{d\mid n}{\mathrm{\Phi }}_d=t^n-1$

for every $n\ge 1$, is discharged by The recursion defines a unique monic ${\mathrm{\Phi }}_n\in \mathbb{Z}[t]$, of degree $\phi (n)$ ↗, which is why that theorem is a numbered result and not a parenthesis: the division is carried out over $\mathbb{Z}$, not over a field, so exactness is a statement about integer coefficients and does not follow from the division algorithm.

• Why not define ${\mathrm{\Phi }}_n$ by its roots. The usual definition takes ${\mathrm{\Phi }}_n$ to be the monic polynomial whose roots are the primitive $n$-th roots of unity in $\mathbb{C}$, or the minimal polynomial over $\mathbb{Q}$ of a primitive $n$-th root of unity. Neither is available at this point in the reading order, and neither would serve: the minimal-polynomial version makes ${\mathrm{\Phi }}_n$ is irreducible in $\mathbb{Q}[t]$ for every $n\ge 1$ a tautology rather than a theorem, and over a base field where the reduction of ${\mathrm{\Phi }}_n$ is reducible it no longer controls the factorisation there (For $\gcd (n,q)=1$ the reduction of ${\mathrm{\Phi }}_n$ in ${\mathbb{F}}_q[t]$ is a product of distinct monic irreducibles, each of degree the order of $[q]$ modulo $n$). The recursion above is a statement about $\mathbb{Z}[t]$ alone, and the description of the roots is then a theorem (Over a field whose characteristic does not divide $n$, the roots of ${\mathrm{\Phi }}_n$ are exactly the primitive roots of unity).

• Reduction into another field. For a field $K$ the image of ${\mathrm{\Phi }}_n$ under the coefficientwise ring homomorphism $\mathbb{Z}[t]\to K[t]$ induced by the canonical map $\mathbb{Z}\to K$ and fixing $t$ is again written ${\mathrm{\Phi }}_n$ when no confusion arises; the identity ${\prod }_{d\mid n}{\mathrm{\Phi }}_d=t^n-1$ is preserved, since a ring homomorphism preserves finite products.