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12(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/06/21(日)08:09 ID:W0WIc7wX(6/18) AAS
>>11
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・チェボタレフの密度定理：
外部ﾘﾝｸ:en.wikipedia.org
Chebotarev's density theorem
Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field {\displaystyle \mathbb {Q} of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime p in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes p less than a large integer N, tends to a certain limit as N goes to infinity. It was proved by Nikolai Chebotaryov in his thesis in 1922, published in (Tschebotareff 1926).
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13: 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/06/21(日)08:09 ID:W0WIc7wX(7/18) AAS
>>12
つづき

Important consequences
The Chebotarev density theorem reduces the problem of classifying Galois extensions of a number field to that of describing the splitting of primes in extensions. Specifically, it implies that as a Galois extension of K, L is uniquely determined by the set of primes of K that split completely in it.[6] A related corollary is that if almost all prime ideals of K split completely in L, then in fact L = K.[7]
外部ﾘﾝｸ:tsujimotterはてなぶろぐ/entry/how-to-use-chebotarev-density-theorem
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2018-12-13
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