Every finite subgroup of the unit group of an integral domain is cyclic

Given: An integral domain $D$ and a finite subgroup $G\le D^{\times }$.

An integral domain is a commutative ring with no zero divisors and with $0\mathrm{\ne }1$ (Zero divisor, and integral domain: a commutative ring with $1\mathrm{\ne }0$ and no zero divisors).

The units $D^{\times }$ form a group under multiplication (The units of a ring are the invertible elements of its multiplicative monoid, and $R^{\times }$ is a group under multiplication; $0\in R^{\times }$ only in the zero ring).

A subgroup contains the identity and is closed under multiplication and inverses (Subgroup).

A group is cyclic when it equals $⟨g⟩$ for some element $g$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups).

A finite set has a natural-number cardinality $\mathrm{\mid }G\mathrm{\mid }$ (The cardinality $\mid A\mid$ of a finite set).

The exponent $e$ of a finite group is the least positive integer such that $g^e=1$ for every group element $g$ (The exponent of a finite group).

If $G\cong C_{n_1}\times \cdots \times C_{n_r}$ with $1<n_1\mid \cdots \mid n_r$, then $\mathrm{\mid }G\mathrm{\mid }=n_1\cdots n_r$ and $\exp (G)=n_r$ (Invariant factors determine the order and exponent of a finite abelian group).

A nontrivial finite abelian group is cyclic exactly when its invariant-factor list has one entry (A nontrivial finite abelian group is cyclic if and only if it has one invariant factor).

A monic polynomial has leading coefficient $1$, and a nonzero polynomial has a natural-number degree (Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree).

Evaluation substitutes a ring element into a formal polynomial, and a root is an element with value zero (Evaluation and roots of a polynomial in a commutative target ring).

A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots (A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots).

Every finite abelian group has a unique invariant-factor list $1<n_1\mid \cdots \mid n_r$, with the trivial group corresponding to the empty list (Fundamental theorem of finite abelian groups: invariant-factor form).