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'table of some fundamental units'
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| Title of object: |
table of some fundamental units |
| Canonical Name: |
SomethingRelatedToFundamentalUnits |
| Type: |
Result |
| Created on: |
2008-04-07 14:32:08 |
| Modified on: |
2008-04-07 14:33:43 |
| Classification: |
msc:11R04, msc:11R11, msc:11R27 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
Below, we tabulate the fundamental units $\eta$ of first real quadratic fields $\mathbb{Q}(\sqrt{d})$; the number $\omega$ is $\sqrt{d}$ for\, $d \not\equiv 1 \pmod{4}$\, and $\displaystyle\frac{1\!+\!\sqrt{d}}{2}$ for\,
$d \equiv 1 \pmod{4}$.
\begin{center}
\begin{tabular}{|c|c|}
\hline\hline
$d$ & $\eta$ \\
\hline\hline
$2$ & $1+\omega$ \\
\hline
$3$ & $2+\omega$ \\
\hline
$5$ & $\omega$ \\
\hline
$6$ & $5+2\omega$ \\
\hline
$7$ & $8+3\omega$ \\
\hline
$10$ & $3+\omega$ \\
\hline
$11$ & $10+3\omega$ \\
\hline
$13$ & $1+\omega$ \\
\hline
$14$ & $15+4\omega$ \\
\hline
$15$ & $4+\omega$ \\
\hline
$17$ & $3+2\omega$ \\
\hline
$19$ & $170+39\omega$ \\
\hline
$21$ & $2+\omega$ \\
\hline
$22$ & $197+42\omega$ \\
\hline
$23$ & $24+5\omega$ \\
\hline
$26$ & $5+\omega$ \\
\hline
$29$ & $2+\omega$ \\
\hline
$30$ & $11+2\omega$ \\
\hline
$31$ & $1520+273\omega$ \\
\hline
$33$ & $19+8\omega$ \\
\hline
$34$ & $35+6\omega$ \\
\hline
$35$ & $6+\omega$ \\
\hline
$37$ & $5+2\omega$ \\
\hline
$38$ & $37+6\omega$ \\
\hline
$39$ & $25+4\omega$ \\
\hline
$41$ & $27+10\omega$ \\
\hline
$42$ & $13+2\omega$ \\
\hline
$43$ & $3482+531\omega$ \\
\hline
$46$ & $24335+3588\omega$ \\
\hline
\end{tabular}
\end{center}
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