現代数学の系譜工学物理雑談古典ガロア理論も読む60

現代数学の系譜工学物理雑談古典ガロア理論も読む60at MATH

現代数学の系譜工学物理雑談古典ガロア理論も読む60 - 暇つぶし2ch315:ﾄy)?…?(x + ζn???1y). Here x and y are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Q(ζn). If unique factorization of algebraic integers were true, then it could have been used to rule out the existence of nontrivial solutions to Fermat's equation. Kummer found a way around this difficulty. He introduced a replacement for the prime numbers in the cyclotomic field Q(ζp), expressed the failure of unique factorization quantitatively via the class number hp and proved that if hp is not divisible by p (such numbers p are called regular primes) then Fermat's theorem is true for the exponent n = p. Furthermore, he gave a criterion to determine which primes are regular and using it, established Fermat's theorem for all prime exponents p less than 100, with the exception of the irregular primes 37, 59, and 67. Kummer's work on the congruences for the class numbers of cyclotomic fields was generalized in the twentieth century by Iwasawa in Iwasawa theory and by Kubota and Leopoldt in their theory of p-adic zeta functions. つづく

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