For what it is worth, I reproduce below the relevant paragraphs in Chowla's aforementioned paper.
The number of solutions of the equation
$$\epsilon - \epsilon^{\prime} =1 \qquad \qquad \qquad(1)$$
where $\epsilon$ and $\epsilon^{\prime}$ are units of a fixed algebraic number field $R(\theta)$ is finite.
Proof. Let $n$ be the degree of the field $R(\theta)$. With the usual notation we write $n=r_{1}+2r_{2}$, $r=r_{1}+r_{2}-1$ (see e.g. the Chelsea reprint of Landau's Algebraische Zahlen) and (in accordance with the Dirichlet-Minkowski theorem)
$$\epsilon = \rho\eta_{1}^{a_{1}} \cdots \eta_{r}^{a_{r}}, \quad \epsilon^{\prime} = \rho^{\prime} \eta_{1}^{a_{1}^{\prime}} \cdots \eta_{r}^{a_{r}^{\prime}}$$
where $\eta_{1}, \ldots, \eta_{r}$ is a system of fundamental units of $R(\theta)$ and $\rho, \rho^{\prime}$ are certain roots of unity belonging to $R(\theta)$. Write for $m\in\{1,\ldots, r\}$
$$a_{m} = q_{m}(2n+1)+t_{m}, \quad a_{m}^{\prime} = q_{m}^{\prime}(2n+1)+t_{m}^{\prime}$$
where the $q$'s and $t$'s are rational integers and $0\leq t_{m}, t_{m}^{\prime} \leq 2n$. Then $(1)$ becomes
$$\rho \, \eta_{t_{1}}\cdots \eta_{r}^{t_{r}} \alpha^{2n+1}-\rho^{\prime} \, \eta_{1}^{t_{1}^{\prime}} \cdots \eta_{r}^{t_{r}^{\prime}} \beta^{2n+1}=1$$
where $\alpha, \beta$ are integers (in fact, units) of $R(\theta)$. By a well-known extension of the Thue-Siegel-Roth theorem (see, for example, vol. 2 of LeVeque's "Topics in number theory", pp. 150-154) the equation
$$\lambda \alpha^{2n+1} - \mu\beta^{2n+1} = 1$$
where $\lambda, \mu$ are fixed integers of $R(\theta)$ and $\alpha, \beta$ are the unknowns (also integers of $R(\theta)$) has only a finite numbers of solutions. Hence our assertion regarding $(1)$.
[Note. $\rho$ and $\rho^{\prime}$ can assume only a finite number of values.]
