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:
“Mine was, as it were, the connecting link between wild and cultivated fields; as some states are civilized, and others half-civilized, and others savage or barbarous, so my field was, though not in a bad sense, a half-cultivated field. They were beans cheerfully returning to their wild and primitive state that I cultivated, and my hoe played the Ranz des Vaches for them.”
—Henry David Thoreau (1817–1862)