...Analytical number theory was developed as a result of the development of Dirichlet's method. Kummer, who followed Dirichlet, started from the study of Fermat's Last Theorem and studied in detail the cyclotomic field generated by the roots of x n -1 = 0. In this field, the uniqueness of prime factorization does not hold, which made research difficult, but Kummer overcame this by considering ideal numbers. ... From [Number Theory of Algebraic Number Fields]…The number theory of algebraic number fields began when CF Gauss considered number theory in the field Q ( i ) (where i is the imaginary unit) known as the Gaussian number field in his research on quadratic remainders. It was later refined by EE Kummer's research on cyclotomic fields and JW Dedekind's theory of ideals, and its foundations were established in D. Hilbert's paper Zahlbericht. Hilbert's concept of class field theory also played an important role in the subsequent development of algebraic number fields. … From Fermat's Last Theorem…Many attempts were made after that, but the most important contribution was made by EE Kummer. The field obtained by adding a root of a power of 1 to the field of rational numbers is called a cyclotomic field. Kummer found a connection between this problem and the number theory of cyclotomic fields, and showed many important results. One of them was that if a prime number p does not divide the numerator of the Bernoulli numbers B 2 , B 4 , …, B p -1 , then the theorem is true for n = p . … *Some of the terminology explanations that mention "cyclotomic field" are listed below. Source | Heibonsha World Encyclopedia 2nd Edition | Information |
…ディリクレの方法を発展させていく中から解析的整数論ができてきた。ディリクレに引き続いて現れたクンマーは,フェルマーの大定理の研究から出発して,xn-1=0の根で生成される円分体を詳しく研究した。この体では,素因数分解の一意性が成り立たず,これが研究を困難にしていたが,クンマーは理想数というものを考えてこれを克服した。… 【代数体の整数論】より…代数体の整数論は,C.F.ガウスが4乗剰余の研究の中でガウス数体と呼ばれる体Q(i)(iは虚数単位)における整数論を考えたことに始まる。その後,E.E.クンマーによる円分体の研究や,J.W.デデキントによるイデアルの理論などによって形が整えられ,D.ヒルベルトの報文《Zahlbericht》においてその基礎が確立した。またヒルベルトが提出した類体論の構想は,その後の代数体の発展に重要な役割を果たした。… 【フェルマーの大定理】より…その後,多くの試みがなされたが,もっとも重要な寄与をしたのは,E.E.クンマーである。有理数体に1のべき根を添加して得られる体を円分体というが,クンマーはこの問題と円分体の整数論との関係を見いだし,多くの重要な結果を示した。その一つとして,素数pがベルヌーイ数B2,B4,……,Bp-1の分子を割らなければ,n=pについて定理は正しいことを示した。… ※「円分体」について言及している用語解説の一部を掲載しています。 出典|株式会社平凡社世界大百科事典 第2版について | 情報 |
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