Kronecker root-of-unity criterion

Given: A number field $K$ of degree $n=[K:\mathbb{Q}]$, the set $\mathrm{\Sigma }={\mathrm{Hom}}_{\mathbb{Q}}(K,\mathbb{C})$ of its $n$ embeddings into $\mathbb{C}$, and an element $0\mathrm{\ne }\alpha \in {\mathcal{O}}_K$ with $\mathrm{\mid }\sigma (\alpha )\mathrm{\mid }\le 1$ for every $\sigma \in \mathrm{\Sigma }$.

$K\mathrm{/}\mathbb{Q}$ is separable, because $\mathbb{Q}$ has characteristic zero, hence is perfect, and algebraic extensions of perfect fields are separable (Fields of characteristic zero, finite fields, and algebraically closed fields are perfect, Every algebraic extension of a perfect field is separable); so the norm is the product over the distinct embeddings, $N_{K\mathrm{/}\mathbb{Q}}(\alpha )={\prod }_{\sigma \in \mathrm{\Sigma }}\sigma (\alpha )$, with $\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }=n=[K:\mathbb{Q}]$ (Norm and trace from embeddings, with the inseparable exponent in the norm formula, Archimedean embeddings and signature).

For $\alpha \in {\mathcal{O}}_K$ the norm $N_{K\mathrm{/}\mathbb{Q}}(\alpha )$ is an integer (Trace and norm of an algebraic integer); if $\alpha \mathrm{\ne }0$ then multiplication by $\alpha$ is an invertible linear map, so $N_{K\mathrm{/}\mathbb{Q}}(\alpha )\mathrm{\ne }0$ (The norm $N_{K\mathrm{/}F}$ and trace ${\mathrm{Tr}}_{K\mathrm{/}F}$ of a finite field extension, Ring of integers).

An element $\beta$ is conjugate to $\alpha$ over $\mathbb{Q}$ exactly when $\beta$ is a complex root of the minimal polynomial $m_{\alpha }\in \mathbb{Q}[X]$ (Conjugate algebraic elements over a field). Sending an embedding $\tau :\mathbb{Q}(\alpha )\to \mathbb{C}$ to $\tau (\alpha )$ is a bijection onto the set of distinct complex roots of $m_{\alpha }$ ($F$-embeddings of $F(\alpha )$ into an algebraically closed field correspond to the distinct roots of $m_{\alpha }$); restriction $\mathrm{\Sigma }\to {\mathrm{Hom}}_{\mathbb{Q}}(\mathbb{Q}(\alpha ),\mathbb{C})$ is surjective (Restriction partitions embeddings in a finite tower into extension fibres); and for $\sigma \in \mathrm{\Sigma }$ one has $m_{\alpha }(\sigma (\alpha ))=\sigma (m_{\alpha }(\alpha ))=0$. Hence the set of complex roots of $m_{\alpha }$ is exactly $\{\sigma (\alpha ):\sigma \in \mathrm{\Sigma }\}$.

Complex modulus is multiplicative, $\mathrm{\mid }zw\mathrm{\mid }=\mathrm{\mid }z\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }$, 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).

Sums and products of elements integral over $\mathbb{Z}$ are integral (Integral elements over a nonzero base ring form a subring), so for $m\ge 1$ the power ${\alpha }^m$ is a nonzero element of ${\mathcal{O}}_K$, the integral closure of $\mathbb{Z}$ in $K$ (Ring of integers).

For every $m\ge 1$ the minimal polynomial $m_{{\alpha }^m}$ of the algebraic element ${\alpha }^m$ has coefficients in $\mathbb{Z}$, and $\mathrm{deg}{m}_{{\alpha }^m}=[\mathbb{Q}({\alpha }^m):\mathbb{Q}]$ divides $[K:\mathbb{Q}]=n$ (Minimal-polynomial criterion for algebraic integers, An element is algebraic over $F$ if and only if its simple extension $F(a)\mathrm{/}F$ is finite, The degree of an intermediate field divides the degree of a finite extension).

Every complex root $w$ of $m_{{\alpha }^m}$ is the image of ${\alpha }^m$ under a $\mathbb{Q}$-embedding of $\mathbb{Q}({\alpha }^m)$ into $\mathbb{C}$ ($F$-embeddings of $F(\alpha )$ into an algebraically closed field correspond to the distinct roots of $m_{\alpha }$). Such an embedding extends to a $\mathbb{Q}$-embedding $\sigma :K\to \mathbb{C}$ (Restriction partitions embeddings in a finite tower into extension fibres), so $w=\sigma (\alpha {)}^m$.

For fixed $n\ge 1$ and $R=1$ there are only finitely many monic integer polynomials of degree at most $n$ whose complex roots, counted with multiplicity, all have modulus at most $1$ (Bounded roots give finitely many monic integer polynomials); this is the supplier consumed here, and the exact obligation used is this instance $R=1$.

A nonzero polynomial of degree at most $n$ over the integral domain $\mathbb{C}$ has at most $n$ distinct roots (A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots).

An element $\zeta$ of $K$ is a root of unity exactly when ${\zeta }^N=1$ 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).