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regular prime
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(Definition)
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A prime  is regular if the class number of the cyclotomic field
 is not divisible by  (where
 denotes a primitive
 root of unity). An irregular prime is a prime that is not regular.
Regular primes rose to prominence as a result of Ernst Kummer's work in the 1850's on Fermat's Last Theorem. Kummer was able to prove Fermat's Last Theorem in the case where the exponent is a regular prime, a result that prior to Wiles's recent work was the only demonstration of Fermat's Last Theorem for a large class of exponents. In the course of this work Kummer also established the following numerical criterion for determining whether a prime is regular:
Based on this criterion it is possible to give a heuristic argument that the regular primes have density  in the set of all primes [1]. Despite this, there is no known proof that the set of regular primes is infinite, although it is known that there are infinitely many irregular primes.
- 1
- Kenneth Ireland & Michael Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, Second Edition, 1990.
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"regular prime" is owned by djao.
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(view preamble)
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irregular prime |
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Cross-references: infinite, proof, density, argument, multiple, Bernoulli numbers, numerators, class, exponent, Fermat's last theorem, root of unity, primitive, divisible, cyclotomic field, class number, regular, prime
There are 5 references to this entry.
This is version 3 of regular prime, born on 2002-06-05, modified 2005-04-20.
Object id is 3040, canonical name is RegularPrime.
Accessed 4670 times total.
Classification:
| AMS MSC: | 11R18 (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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Pending Errata and Addenda
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