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How do I go about finding an epsilon-approximation of a given real number x by an element of minimal or near-minimal height from a given number field K?

It would suffice for my purposes to solve this for real quadratic extensions of Q.

Case of primary interest is actually K = Q(sqrt(2)).

Thanks. Alex--

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This can be done by the LLL algorithm. (See, e.g., http://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm#Applications and http://en.wikipedia.org/wiki/Integer_relation_algorithm)

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