Cyclotomic fields are fields obtained by adjoining to ℚ roots of unity, i.e. roots of polynomials of the form Xn - 1, although the reader is warned that this terminology will be extended in §2. Geometrically these arise from the problem of dividing the unit circle into equal parts. They form a class of algebraic number fields with very beautiful and important properties. Cyclotomic fields play a fundamental role in a number of arithmetic problems: for instance primes in arithmetic progressions (see VIII,§4) and Fermat's Last Theorem (see VII,§1). They also throw new light on the theory of quadratic fields which we have already considered in (V,§1), and in particular provide the background to the “most natural” proof of the quadratic reciprocity law. They are Galois extensions of ℚ with abelian Galois group, and conversely, as one can show, although we shall not be able to, any finite extension of ℚ with abelian Galois group is a subfield of a cyclotomic field. Thus the theory looks forward to one of the most sophisticated aspects of algebraic number theory: the so-called class field theory.

Basic theory

We begin by considering some field theoretic aspects, where the base field is not necessarily the field of rationals. Observe that any finite subgroup S of the multiplicative group of a field is necessarily cyclic: for, in a non-cyclic finite abelian group G of exponent h, there are |G| > h roots of Xh = 1G, while in a field the equation Xh - 1 has at most h solutions.