units of real cubic fields with exactly one real embedding

Let $K\subseteq ℝ$ be a number field with $[K:\mathbb{Q}]=3$ such that $K$ has exactly one real embedding. Thus, $r=1$ and $s=1$. Let $𝒪_K{}^{*}$ denote the group of units of the ring of integers of $K$. By Dirichlet’s unit theorem, $𝒪_K{}^{*}\cong \mu (K)\times \mathbb{Z}$ since $r+s-1=1$. The only roots of unity in $K$ are $1$ and $-1$ because $K\subseteq ℝ$. Thus, $\mu (K)=\{1,-1\}$. Therefore, there exists $u\in 𝒪_K{}^{*}$ with $u>1$, such that every element of $𝒪_K{}^{*}$ is of the form $\pm u^n$ for some $n\in \mathbb{Z}$.

Let $\rho >0$ and 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, $u={\displaystyle \frac{1}{{\rho }^2}}$. One can then deduce that $\mathrm{disc}u=-4{\sin }^2\theta {\left({\rho }^3+{\displaystyle \frac{1}{{\rho }^3}}-2\cos \theta \right)}^2$. 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 $|\mathrm{disc}u|\le 4\left(u^3+{\displaystyle \frac{1}{u^3}}+4\right)$. Define $d=|\mathrm{disc}{𝒪}_K|$. Then $d\le |\mathrm{disc}u|\le 4\left(u^3+{\displaystyle \frac{1}{u^3}}+4\right)$. Thus, $u^3\ge {\displaystyle \frac{d}{4}}-4-{\displaystyle \frac{1}{u^3}}$. From this, one can obtain that $u^3\ge {\displaystyle \frac{d-16+\sqrt{d^2-32d+192}}{8}}$. (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, $u^2\ge {\left({\displaystyle \frac{d-16+\sqrt{d^2-32d+192}}{8}}\right)}^{\frac{2}{3}}$. Therefore, if an element $x\in 𝒪_K{}^{*}$ can be found such that , 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 ${\left({\displaystyle \frac{23+10\sqrt{21}}{2}}\right)}^{\frac{2}{3}}$, which is larger than $9$. Note that $\left(\sqrt[3]{4}+\sqrt[3]{2}+1\right)\left(\sqrt[3]{2}-1\right)=2-1=1$. Since , it follows that $\sqrt[3]{4}+\sqrt[3]{2}+1$ is the fundamental unit of ${𝒪}_K$.

• 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 $K$.

Title	units of real cubic fields with exactly one real embedding
Canonical name	UnitsOfRealCubicFieldsWithExactlyOneRealEmbedding
Date of creation	2013-03-22 16:02:25
Last modified on	2013-03-22 16:02:25
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	13
Author	Wkbj79 (1863)
Entry type	Application
Classification	msc 11R27
Classification	msc 11R16
Classification	msc 11R04
Related topic	NormAndTraceOfAlgebraicNumber