A set in \(\mathbb{R}^n\) is called Salem if it supports a probability measure whose Fourier transform decays as fast as the Hausdorff dimension of the set will allow. We construct the first explicit (i.e., non-random) examples of Salem sets in \(\mathbb{R}^n\) of arbitrary prescribed Hausdorff dimension. This completely resolves a problem proposed by Kahane more than 60 years ago. The construction is based on a form of Diophantine approximation in number fields. This is joint work with Robert Fraser.