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I was thinking given a finite extension of Q, call it K... find a galois extension that includes the finite extension, call it L... then somehow use the fact that the ring of algebraic integers of L has an integral basis to show that the integers of K must have an integral basis.

By integral basis, I mean that there is a finite set of elements in the ring w1,w2,w3...wn such that an element in the ring can be written in the form a1w1+a2w2+a3w3+...anwn

where a1,a2,a3,...an belong to Z.

I'd appreciate any help. Thanks.