examples of regular primesExampleOfRegularPrime2003-12-19 14:55:492005-03-10 14:02:53Exampleregular primeclass numbercyclotomic field% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Rats}{\mathbb{Q}}{\bf Examples}:
\begin{enumerate}
\item These are all the irregular primes up to $1061$:
37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271,\\
283, 293, 307, 311, 347, 353, 379, 389, 401,\\
409, 421, 433, 461, 463, 467, 491, 523, 541,\\
547, 557, 577, 587, 593, 607, 613, 617, 619,\\
631, 647, 653, 659, 673, 677, 683, 691, 727,\\
751, 757, 761, 773, 797, 809, 811, 821, 827,\\
839, 877, 881, 887, 929, 953, 971, 1061.
(for this, see the \PMlinkexternal{On-Line Encyclopedia of Integer Sequences}{http://www.research.att.com/~njas/sequences/Seis.html}, \PMlinkexternal{
sequence A000928}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.c
gi?Anum=A000928})
\item The following are the first few class numbers of the cyclotomic fields $\Rats(\zeta_p)$, where $\zeta_p$ is a primitive $p$-th root of unity:
\begin{tabular}{|c|c|}
\hline
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
$p$ & Class Number \\
\hline
3 & 1 \\
5 & 1 \\
7 & 1 \\
11 & 1 \\
13 & 1 \\
17 & 1 \\
19 & 1 \\
23 & 3 \\
29 & 8 \\
31 & 9 \\
37 & 37 \\
41 & 121 \\
43 & 211 \\
47 & 695 \\
53 & 4889 \\
59 & 41241 \\
61 & 76301 \\
\hline
\end{tabular}
An excellent reference for this is $\cite{wash}$.
{\bf Remarks}:
\begin{itemize}
\item Notice that $37$ divides $37$, and $59$ divides $41241=3\cdot 59\cdot 233$, thus $37,\ 59$ are irregular primes (see above).
\item The class number of the cyclotomic fields grows very quickly with $p$. For example, $p=19$ is the last cyclotomic field of class number 1.
\end{itemize}
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{wash} L. C. Washington, {\em Introduction to Cyclotomic Fields},
Springer-Verlag, New York.
\end{thebibliography}