units of quadratic fieldsUnitsOfQuadraticFields2004-03-21 18:20:052008-04-09 11:28:02ApplicationDirichlet's unit theorem% this is the default PlanetMath preamble. as your knowledge
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Dirichlet's unit theorem gives all units of an algebraic number field $\mathbb{Q}(\vartheta)$ in the unique form
$$\varepsilon = \zeta^{n}\eta_1^{k_1}\eta_2^{k_2}...\eta_t^{k_t},$$
where $\zeta$ is a primitive $w^\mathrm{th}$ root of unity in $\mathbb{Q}(\vartheta)$, the $\eta_j$'s are the fundamental units of $\mathbb{Q}(\vartheta)$,\, $0 \leqq n \leqq w\!-\!1$,\, $k_j \in \mathbb{Z}$\, $\forall j$,\, $t = r\!+\!s\!-\!1$.
\begin{itemize}
\item The case of a real quadratic field $\mathbb{Q}(\sqrt{m})$, the square-free \,$m > 1$:\, $r = 2$,\, $s = 0$,\, $t = r\!+\!s\!-\!1 = 1$.\, So we obtain
$$\varepsilon = \zeta^{n}\eta^{k} = \pm\eta^{k},$$
because\, $\zeta= -1$\, is the only real primitive 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 any imaginary quadratic field $\mathbb{Q}(\vartheta)$; here\, $\vartheta = \sqrt{m}$,\, the square-free\, $m < 0$:\, The conjugates of $\vartheta$ are the pure imaginary numbers $\pm\sqrt{m}$, hence\, $r = 0$,\, $2s = 2$,\, $t = r\!+\!s\!-\!1 = 0$.\, Thus we see that all units are
$$\varepsilon = \zeta^{n}.$$
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 \PMlinkname{cyclotomic field}{CyclotomicExtension} 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\, $\varepsilon = (\frac{1+\sqrt{-3}}{2})^{n}$,\, where\, $n = 0,\,1,\,\ldots,\,5$, or, equivalently,
\, $\varepsilon = \pm(\frac{-1+\sqrt{-3}}{2})^{n}$,\, 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\,
$(-1)^{n}$, where\, $n = 0,\,1$.
\end{itemize}