Kronecker–Weber theorem

In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form ${\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}$. The Kronecker–Weber theorem provides a partial converse: every finite abelian extension of Q is contained within some cyclotomic field. In other words, every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients. For example,

${\displaystyle {\sqrt {5}}=e^{2\pi i/5}-e^{4\pi i/5}-e^{6\pi i/5}+e^{8\pi i/5},}$ ${\displaystyle {\sqrt {-3}}=e^{2\pi i/3}-e^{4\pi i/3},}$ and ${\displaystyle {\sqrt {3}}=e^{\pi i/6}-e^{5\pi i/6}.}$

The theorem is named after Leopold Kronecker and Heinrich Martin Weber.