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In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to , the field of rational numbers. The -th cyclotomic field (where ) is obtained by adjoining a primitive -th root of unity to the rational numbers. The cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's last theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime ) – and more precisely, because of the failure of unique factorization in their rings of integers – that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences. == Properties == A cyclotomic field is the splitting field of the cyclotomic polynomial : and therefore it is a Galois extension of the field of rational numbers. The degree of the extension : is given by where is Euler's phi function. A complete set of Galois conjugates is given by , where runs over the set of invertible residues modulo (so that is relative prime to ). The Galois group is naturally isomorphic to the multiplicative group : of invertible residues modulo , and it acts on the primitive th roots of unity by the formula : . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cyclotomic field」の詳細全文を読む スポンサード リンク
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