Hilbert-Speiser Theorem

The Hilbert-Speiser theorem is a result in algebraic number theory characterizing which cyclotomic fields possess a normal integral basis; more generally it applies to any finite abelian extension of the rational numbers, since by the Kronecker-Weber theorem every such extension is isomorphic to a subfield of a cyclotomic field. The theorem states that a finite abelian extension of the rationals has a normal integral basis if and only if it is tamely ramified over the rationals, equivalent to being a subfield of the field generated by the nth roots of unity for some squarefree odd number n. The result was introduced by David Hilbert in 1897 and by Andreas Speiser in 1916; a converse was later proved by Cornelius Greither, Daniel Replogle and Karl Rubin among others in 1999. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/