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,

${\displaystyle {\sqrt {5}}=e^{2\pi i/5}-e^{4\pi i/5}-e^{6\pi i/5}+e^{8\pi i/5},}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sqrt{-3} = e^{2 \pi i / 3} - e^{4 \pi i / 3},} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sqrt{3} = e^{\pi i / 6} - e^{5 \pi i / 6}.}

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