We prove a splitting theorem for the n-r.e. degrees, of which the Sacks Splitting Theorem [9] for the r.e. (= 1-r.e.) degrees is a special case. For background terminology and notation see [4] and [11] . Interest in the n-r.e. (and more particularly, d-r.e. = 2-r.e.) degrees stems from their affinity even for large n with the r.e. degrees, and a number of recent papers (see for example [1, 2, 4, 6] ) have sought to clarify to what extent there are similarities and differences at the individual

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... evels of the n-r.e. degree hierarchy. Expectations that the few global properties of the r.e. degrees (in particular density) might carry over to the n-r.e. degrees, n > 1, have been only partially fulfilled. Lachlan observed that n-r.e. degrees are not minimal in the n-r.e. degrees, essentially because they are n-REA (see [7] ), but density fails, even under 0' (see [3] ). On the other hand, some of the pathological properties of the r.e. degrees disappear in the wider context, such as in Downey's diamond theorem [5] for the d-r.e. degrees and Arslanov's use [1] (and more generally in [4]), of d-r.e. degrees for cupping. Therefore the theorem below is welcome in the sense that we now have a global splitting property at every level of the n-r.e. hierarchy of degrees. (We note that the splitting of the n-r.e. degrees in the A2 degrees is an immediate consequence of the n-REA property and the relativised Sacks Splitting Theorem.) Further information relating to the structure of the n-r.e. degrees may be found in [8] . Theorem. Let d be n-r.e. Then there exist n-r.e. degrees a, b such that a|b and aub = d. In order to prove the theorem, we give a construction for the case d properly d-r.e. and then indicate how our construction can be adapted to give the inductive step in the proof of the theorem for all « > 1 . The case zz = 1 is just the Sacks Splitting Theorem, of course. Let D G d be d-r.e. We construct d-r.e. sets A, B, and partial recursive (p.r.) functionals T, A, and Q, satisfying the overall requirements A = YD, B = AD, D = QAB