  We prove an analogue of a result by Goldston, Pintz and Yildirim for small
gaps between primes that split completely in an abelian number field. We prove
both a conditional result assuming the Elliott-Halberstam conjecture, and an
unconditional result. We also give another proof of the same result in the
special case of a quadratic extension of class number 1, which relies on a
generalization of the Bombieri-Vinogradov theorem for quadratic number fields.
