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 $r+s-1=1$. 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 $4\sin^{2}\theta(x-2\cos\theta)^{2}-4x^{2}$ 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 $\displaystyle u^{3}\geq\frac{d-16+\sqrt{d^{2}-32d+192}}{8}$. (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, $\displaystyle u^{2}\geq\left(\frac{d-16+\sqrt{d^{2}-32d+192}}{8}\right)^{\frac% {2}{3}}$. Therefore, if an element $x\in{\mathcal{O}_{K}}^{*}$ can be found such that $\displaystyle 1, 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 $\displaystyle\left(\sqrt[3]{4}+\sqrt[3]{2}+1\right)\left(\sqrt[3]{2}-1\right)=% 2-1=1$. 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: Springer-Verlag, 1977.
Title units of real cubic fields with exactly one real embedding UnitsOfRealCubicFieldsWithExactlyOneRealEmbedding 2013-03-22 16:02:25 2013-03-22 16:02:25 Wkbj79 (1863) Wkbj79 (1863) 13 Wkbj79 (1863) Application msc 11R27 msc 11R16 msc 11R04 NormAndTraceOfAlgebraicNumber