[nkurz]

Based on your other comments, you probably already understand this correctly, but for others who might be confused, the actual policy is not that it goes on repeated vowels, but that it goes on the second vowel in a pair to make clear that it starts another syllable. Under this policy, coöperate and naïve get diaereses, but books never would (unless you were somehow trying to indicate a special case where it was pronounced with two syllables). Here's more: https://en.wikipedia.org/wiki/Diaeresis_(diacritic)