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[parent] units of real cubic fields with exactly one real embedding (Application)

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 \vert\operatorname{disc}u\vert \le 4\left(u^3+\frac{1}{u^3}+4\right)$. Define $ d=\vert\operatorname{disc}\mathcal{O}_K\vert$. Then $ \displaystyle d\le\vert\operatorname{disc}u\vert \le 4\left(u^3+\frac{1}{u^3}+4\right)$. Thus, $ \displaystyle u^3 \ge \frac{d}{4}-4-\frac{1}{u^3}$. From this, one can obtain that $ \displaystyle u^3 \ge \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 \ge \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<x<\left( \frac{d-16+\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)=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$.

Bibliography

1
Marcus, Daniel A. Number Fields. New York: Springer-Verlag, 1977.



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See Also: norm and trace of algebraic number


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Cross-references: pure cubic field, fundamental unit, applications, lower bound, polynomial, unit, conjugates, roots of unity, Dirichlet's unit theorem, ring of integers, group of units, real embedding, number field
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This is version 10 of units of real cubic fields with exactly one real embedding, born on 2006-06-26, modified 2008-02-26.
Object id is 8090, canonical name is UnitsOfRealCubicFieldsWithExactlyOneRealEmbedding.
Accessed 1084 times total.

Classification:
AMS MSC: 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
 11R16 (Number theory :: Algebraic number theory: global fields :: Cubic and quartic extensions)
 11R27 (Number theory :: Algebraic number theory: global fields :: Units and factorization)
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