Finitely many roots of unity in a number field

Given: A number field $K$ of degree $n=[K:\mathbb{Q}]$, and the set $\mu (K)={\bigcup }_{N\ge 1}{\mu }_N(K)$ of the elements of $K$ that satisfy $x^N=1$ for some $N\ge 1$.

For every $N\ge 1$ the set ${\mu }_N(K)=\{x\in K:x^N=1\}$ is a subgroup of $K^{\times }$, and an element $x\in K$ is a root of unity exactly when $x^N=1$ for some $N\ge 1$ (The group ${\mu }_n(K)$ of $n$-th roots of unity in a field, and primitive $n$-th roots of unity).

An element $b$ of a commutative ring $B$ is integral over a subring $A$ when it is a root of a monic polynomial in $A[X]$, and an algebraic integer is a complex number integral over $\mathbb{Z}$ (Integral elements over a commutative ring and algebraic integers); the ring of integers ${\mathcal{O}}_K$ is the integral closure of $\mathbb{Z}$ in $K$ (Ring of integers).

For $\alpha \in K$, one has $\alpha \in {\mathcal{O}}_K$ if and only if the monic minimal polynomial of $\alpha$ over $\mathbb{Q}$ lies in $\mathbb{Z}[X]$ (Minimal-polynomial criterion for algebraic integers).

For an algebraic element $a$ of an extension of $\mathbb{Q}$, the monic minimal polynomial $m_a\in \mathbb{Q}[X]$ satisfies $f(a)=0$ if and only if $m_a\mid f$ in $\mathbb{Q}[X]$ (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element).

Complex modulus satisfies $\mathrm{\mid }zw\mathrm{\mid }=\mathrm{\mid }z\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }$, $\mathrm{\mid }z\mathrm{\mid }\ge 0$ and $\mathrm{\mid }z\mathrm{\mid }=0$ only for $z=0$ (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive); the $N$-th roots of unity in $\mathbb{C}$ are the numbers $e^{2\pi ik\mathrm{/}N}$, $k=0,\dots ,N-1$, all of modulus one (The $n$-th roots of a complex number and the $n$ distinct roots of unity for every $n\ge 1$).

If $a$ is algebraic over $F$ with minimal polynomial of degree $d$, then $[F(a):F]=d$ (An element is algebraic over $F$ if and only if its simple extension $F(a)\mathrm{/}F$ is finite).

If $F\subseteq E\subseteq L$ and $L\mathrm{/}F$ is finite, then $E\mathrm{/}F$ and $L\mathrm{/}E$ are finite and $[E:F]$ divides $[L:F]$ (The degree of an intermediate field divides the degree of a finite extension); for finite extensions the degrees multiply, $[L:F]=[L:E][E:F]$ (Tower law for finite extensions: $[L:F]=[L:K][K:F]$, The degree $[K:F]={\dim }_{F}K$ of a finite field extension).

There are only finitely many monic integer polynomials of degree at most $n$ whose complex roots, counted with multiplicity, all have modulus at most $R=1$ (Bounded roots give finitely many monic integer polynomials). This is the one batch-2 supplier consumed here, authored in this run; the exact obligation used is that the set of monic integer polynomials of degree at most $n$ all of whose complex roots have modulus at most $1$ is finite.

A nonzero polynomial of degree $d$ over an integral domain has at most $d$ distinct roots in that domain (A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots); $\mathbb{C}$ is a field, hence an integral domain.