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Revision difference : units of quadratic fields |
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Version 35 |
| \PMlinkescapeword{primitive} |
\PMlinkescapeword{primitive} |
| Dirichlet's unit theorem gives all units of an algebraic number field $\mathbb{Q}(\vartheta)$ in the unique form |
Dirichlet's unit theorem gives all units of an algebraic number field $\mathbb{Q}(\vartheta)$ in the unique form |
| $$\varepsilon = \zeta^{n}\eta_1^{k_1}\eta_2^{k_2}...\eta_t^{k_t},$$ |
$$\varepsilon = \zeta^{n}\eta_1^{k_1}\eta_2^{k_2}...\eta_t^{k_t},$$ |
| where $\zeta$ is a primitive $w^\mathrm{th}$ root of unity in $\mathbb{Q}(\vartheta)$, the $\eta_j$'s are the fundamental units of $\mathbb{Q}(\vartheta)$,\, $0 \leqq n \leqq w\!-\!1$,\, $k_j \in \mathbb{Z}$\, $\forall j$,\, $t = r\!+\!s\!-\!1$. |
where $\zeta$ is a primitive $w^\mathrm{th}$ root of unity in $\mathbb{Q}(\vartheta)$, the $\eta_j$'s are the fundamental units of $\mathbb{Q}(\vartheta)$,\, $0 \leqq n \leqq w\!-\!1$,\, $k_j \in \mathbb{Z}$\, $\forall j$,\, $t = r\!+\!s\!-\!1$. |
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| \begin{itemize} |
\begin{itemize} |
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\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
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\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
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| $$\varepsilon = \zeta^{n}\eta^{k} = \pm\eta^{k},$$ |
$$\varepsilon = \zeta^{n}\eta^{k} = \pm\eta^{k},$$ |
| 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 $\eta$. |
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 $\eta$. |
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| Examples:\, If\, $m = 3$,\, then\, $\eta = 2\!+\!\sqrt{3}$;\, if\, $m = 421$,\, then\, $\eta = \frac{444939+21685\sqrt{421}}{2}$. |
Examples:\, If\, $m = 3$,\, then\, $\eta = 2\!+\!\sqrt{3}$;\, if\, $m = 421$,\, then\, $\eta = \frac{444939+21685\sqrt{421}}{2}$. |
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| \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 |
\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 |
| $$\varepsilon = \zeta^{n}.$$ |
$$\varepsilon = \zeta^{n}.$$ |
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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\, $n = 0,\,1,\,2,\,3$.
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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\, $n = 0, 1, 2, 3$.
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| 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 |
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 |
| $$\zeta = \cos{\frac{2\pi}{6}}+i\sin{\frac{2\pi}{6}};$$ |
$$\zeta = \cos{\frac{2\pi}{6}}+i\sin{\frac{2\pi}{6}};$$ |
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hence all units are\, $\varepsilon = (\frac{1+\sqrt{-3}}{2})^{n}$,\, where\, $n = 0,\,1,\,\ldots,\,5$, or, equivalently,
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hence all units are\, $\varepsilon = (\frac{1+\sqrt{-3}}{2})^{n}$,\, where\, $n = 0, 1, \ldots, 5$, or, equivalently,
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| \, $\varepsilon = \pm(\frac{-1+\sqrt{-3}}{2})^{n}$,\, where\, |
\, $\varepsilon = \pm(\frac{-1+\sqrt{-3}}{2})^{n}$,\, where\, |
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$n = 0,\,1,\,2$.
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$n = 0, 1, 2$.
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| 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\, |
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\, |
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$(-1)^{n}$, where\, $n = 0,\,1$.
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$(-1)^{n}$, where\, $n = 0, 1$.
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| \end{itemize} |
\end{itemize} |
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