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units of real cubic fields with exactly one real embedding
Let $K\subseteq\mathbb{R}$ be a number field with $[K\!:\!\mathbb{Q}]=3$ such that $K$ has exactly one real embedding. Thus, $r=1$ and $s=1$. Let ${\mathcal{O}_{K}}^{*}$ denote the group of units of the ring of integers of $K$. By Dirichlet’s unit theorem, ${\mathcal{O}_{K}}^{*}\cong\mu(K)\times\mathbb{Z}$ since $r+s1=1$. The only roots of unity in $K$ are $1$ and $1$ because $K\subseteq\mathbb{R}$. Thus, $\mu(K)=\{1,1\}$. Therefore, there exists $u\in{\mathcal{O}_{K}}^{*}$ with $u>1$, such that every element of ${\mathcal{O}_{K}}^{*}$ is of the form $\pm u^{n}$ for some $n\in\mathbb{Z}$.
Let $\rho>0$ and $0<\theta<\pi$ such that the conjugates of $u$ are $\rho e^{{i\theta}}$ and $\rho e^{{i\theta}}$. Since $u$ is a unit, $N(u)=\pm 1$. Thus, $\pm 1=N(u)=u(\rho e^{{i\theta}})(\rho e^{{i\theta}})=u\rho^{2}$. Since $u>0$ and $\rho^{2}>0$, it must be the case that $u\rho^{2}=1$. Thus, $\displaystyle u=\frac{1}{\rho^{2}}$. One can then deduce that $\displaystyle\operatorname{disc}u=4\sin^{2}\theta\left(\rho^{3}+\frac{1}{\rho% ^{3}}2\cos\theta\right)^{2}$. Since the maximum value of the polynomial $4\sin^{2}\theta(x2\cos\theta)^{2}4x^{2}$ is at most $16$, one can deduce that $\displaystyle\operatorname{disc}u\leq 4\left(u^{3}+\frac{1}{u^{3}}+4\right)$. Define $d=\operatorname{disc}\mathcal{O}_{K}$. Then $\displaystyle d\leq\operatorname{disc}u\leq 4\left(u^{3}+\frac{1}{u^{3}}+4\right)$. Thus, $\displaystyle u^{3}\geq\frac{d}{4}4\frac{1}{u^{3}}$. From this, one can obtain that $\displaystyle u^{3}\geq\frac{d16+\sqrt{d^{2}32d+192}}{8}$. (Note that a higher lower bound on $u^{3}$ is desirable, and the one stated here is much higher than that stated in Marcus.) Thus, $\displaystyle u^{2}\geq\left(\frac{d16+\sqrt{d^{2}32d+192}}{8}\right)^{{% \frac{2}{3}}}$. Therefore, if an element $x\in{\mathcal{O}_{K}}^{*}$ can be found such that $\displaystyle 1<x<\left(\frac{d16+\sqrt{d^{2}32d+192}}{8}\right)^{{\frac{2}{% 3}}}$, then $x=u$.
Following are some applications:

The above is most applicable for finding the fundamental unit of a ring of integers of a pure cubic field. For example, if $K=\mathbb{Q}(\sqrt[3]{2})$, then $d=108$, and the lower bound on $u^{2}$ is $\displaystyle\left(\frac{23+10\sqrt{21}}{2}\right)^{{\frac{2}{3}}}$, which is larger than $9$. Note that $\displaystyle\left(\sqrt[3]{4}+\sqrt[3]{2}+1\right)\left(\sqrt[3]{2}1\right)=% 21=1$. Since $1<\sqrt[3]{4}+\sqrt[3]{2}+1<9$, it follows that $\sqrt[3]{4}+\sqrt[3]{2}+1$ is the fundamental unit of $\mathcal{O}_{K}$.

The above can also be used for any number field $K$ with $[K\!:\!\mathbb{Q}]=3$ such that $K$ has exactly one real embedding. Let $\sigma$ be the real embedding. Then the above produces the fundamental unit $u$ of $\sigma(K)$. Thus, $\sigma^{{1}}(u)$ is a fundamental unit of $K$.
References
 1 Marcus, Daniel A. Number Fields. New York: SpringerVerlag, 1977.
Mathematics Subject Classification
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