Cyclotomic fields, fields obtained by adjoining the nth roots of unity to the rational numbers, play a pivotal role in number theory as they contain all abelian extensions of the field of rational numbers according to the Kronecker–Weber theorem. They also offer insights into reciprocity laws and certain Diophantine equations, such as the Fermat equation.

In this chapter, we offer a second proof of the quadratic reciprocity law, which shows how Gauss sums over roots of unity encode quadratic residue behavior. This approach reveals the intrinsic connection between quadratic fields and cyclotomic fields.

We prove the particular case of the Kronecker–Weber theorem asserting that a quadratic field of discriminant D is contained minimally in the cyclotomic field of |D|th roots of unity.

Of great historical relevance is Kummer’s proof of the first case of Fermat’s Last Theorem for regular primes, which uses the unique factorization of ideals in cyclotomic fields. Kummer introduced ideal theory to overcome limitations posed by the non-unique factorization of elements in attempted proofs of FLT. His work laid the foundation for modern algebraic number theory, illustrating the profound utility of cyclotomic methods in solving classical problems and establishing these fields as central objects in arithmetic geometry.