Cyclotomic fields | Algebraic number theory

Cyclotomic field

In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers. 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 n) – 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. (Wikipedia).

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Cyclotomic unit | Ring of integers | Index of a subgroup | Isomorphism | Minimal polynomial (field theory) | Torsion subgroup | Fermat's Last Theorem | Root of unity | Carl Friedrich Gauss | Ideal number | Euler's totient function | Multiplicative group of integers modulo n | Rational number | Kenkichi Iwasawa | Finite group | Serge Lang | Natural transformation | Splitting field | Field extension | Heptadecagon | Kronecker–Weber theorem | Field (mathematics) | Fundamental theorem of arithmetic | Constructible polygon | Integer | Regular prime | Abelian extension | Number theory | Galois group | Prime number | Cyclotomic polynomial | Irreducible polynomial | Discriminant of an algebraic number field | Abstract algebra | Complex number | Subgroup | Ernst Kummer | Iwasawa theory | Galois extension | Finitely generated abelian group | Algebraic integer | Regular polygon | Conjugate element (field theory) | Modular arithmetic | Degree of a field extension