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'units of quadratic fields'
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| Title of object: |
units of quadratic fields |
| Canonical Name: |
UnitsOfQuadraticFields |
| Type: |
Application |
| Created on: |
2004-03-21 18:20:05 |
| Modified on: |
2004-08-06 12:07:17 |
| Classification: |
msc:11R04 |
Preamble:
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Content:
The Dirichlet's unit theorem gives all units of an algebraic number field $\mathbb{Q}(\theta)$ in the unique form
$$\epsilon = \zeta^{n}\eta_1^{k_1}\eta_2^{k_2}...\eta_t^{k_t},$$
where $\zeta$ is a primitive $w$th root of unity in $\mathbb{Q}(\theta)$, the $\eta_j$'s are the so-called {\em fundamental units} of $\mathbb{Q}(\theta)$, \,$0 \le n \le w-1$, \,$k_j \in \mathbb{Z}$ \,$\forall j$, \,$t = r+s-1$.
\begin{itemize}
\item The case of a real quadratic field $\mathbb{Q}(\sqrt{m})$, the square-free $m > 0$: \,$r = 2$, \,$s = 0$, \,$t = r+s-1 = 1$. \,So we obtain
$$\epsilon = \zeta^{n}\eta^{k} = \pm\eta^{k},$$
because \,$\zeta= -1$\, is the only primitive real root of unity ($w =2$). \,Thus, every real quadratic field has infinitely many units and a unique fundamental unit $\eta$.
Examples: \,If \,$m = 3$, then \,$\eta = 2+\sqrt{3}$; if \,$m = 421$, then \,$\eta = \frac{444939+21685\sqrt{421}}{2}$.
\item The case of an imaginary quadratic field $\mathbb{Q}(\theta)$; here \,$\theta = \sqrt{m}$, the square-free $m < 0$: \,The conjugates of $\theta$ are the imaginary numbers $\pm\sqrt{m}$, hence \,$r = 0$, \,$2s = 2$, \,$t = r+s-1 = 0$. \,Thus we see that all units are
$$\epsilon = \zeta^{n}.$$
1) $m = -1$. \,The field contains the primitive fourth root of unity, e.g. $i$, and therefore all units in the {\em \PMlinkescapetext{Gaussian} field} $\mathbb{Q}(i)$ are $i^n$, where \,$n = 0, 1, 2, 3$.
2) $m = -3$. \,The field in question is a {\em cyclotomic field} containing the primitive third root of unity and also the primitive sixth root of unity, namely
$$\zeta = \cos{\frac{2\pi}{6}}+i\sin{\frac{2\pi}{6}};$$
hence all units are
\,$\epsilon = (\frac{1+\sqrt{-3}}{2})^{n}$, where \,$n = 0, 1, ..., 5$,
or, equivalently,
\,$\epsilon = \pm(\frac{-1+\sqrt{-3}}{2})^{n}$, where \,$n = 0, 1, 2$.
3) $m = -2$, $m <-3$. \,The only roots of unity in the field are $\pm 1$; hence \,$\zeta = -1$, \,$w = 2$, and the units of the field are simply
\,$\epsilon = (-1)^{n}$, where \,$n = 0, 1$.
\end{itemize} |
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