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[parent] examples of regular primes (Example)

Examples:

  1. 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 On-Line Encyclopedia of Integer Sequences, sequence A000928)

  2. 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:
    $p$ Class Number
    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

    An excellent reference for this is $\cite{wash}$

    Remarks:

    • Notice that $37$ divides $37$ and $59$ divides $41241=3\cdot 59\cdot 233$ thus $37,\ 59$ are irregular primes (see above).
    • 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.

Bibliography

1
L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.




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See Also: topics on ideal class groups and discriminants

Keywords:  class number, cyclotomic field

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Cross-references: divides, reference, root of unity, primitive, cyclotomic fields, class numbers, irregular primes
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This is version 7 of examples of regular primes, born on 2003-12-19, modified 2005-03-10.
Object id is 5494, canonical name is ExampleOfRegularPrime.
Accessed 2483 times total.

Classification:
AMS MSC11R18 (Number theory :: Algebraic number theory: global fields :: Cyclotomic extensions)
 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants)

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