Fermat’s Last Theorem for Regular Primes
نویسنده
چکیده
For a prime p, we call p regular when the class number hp = h(Q(ζp)) of the pth cyclotomic field is not divisible by p. For instance, all primes p ≤ 19 have hp = 1, so they are regular. Since h23 = 3, 23 is regular. All primes less then 100 are regular except for 37, 59, and 67. By the tables in [6], h37 = 37, h59 = 3 · 59 · 233 and h67 = 67 · 12739. It is known that there are infinitely many irregular primes, and heuristics and tables suggest around 61% of primes should be regular [6, p. 63]. The significance of a prime p being regular is that if the pth power of an ideal a in Z[ζp] is principal, then a is itself principal. Indeed, if ap is principal, then it is trivial in the class group of Q(ζp). Since p doesn’t divide hp, this means a is trivial in the class group, so a is a principal ideal. The concept of regular prime was introduced by Kummer in his work on Fermat’s Last Theorem (FLT). He proved the following, which we will treat in this paper.
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