In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers. The n-th cyclotomic field Q(ζn) (with n > 2) is obtained by adjoining a primitive n-th root of unity ζn 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 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.
Read more about Cyclotomic Field: Properties, Relation With Regular Polygons, Relation With Fermat's Last Theorem
Famous quotes containing the word field:
“The planter, who is Man sent out into the field to gather food, is seldom cheered by any idea of the true dignity of his ministry. He sees his bushel and his cart, and nothing beyond, and sinks into the farmer, instead of Man on the farm.”
—Ralph Waldo Emerson (1803–1882)