Positive rationals appear as squared edge lengths of regular d-simplices in Q^n precisely when the codimension n-d is at least 3 or when they satisfy square-class, norm-group, or Hilbert-symbol conditions for codimensions 0-2.
Beeson, Triangles with vertices on lattice points, The American Mathematical Monthly 99 (1992), no
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Squared edge lengths of regular simplices with rational vertices
Positive rationals appear as squared edge lengths of regular d-simplices in Q^n precisely when the codimension n-d is at least 3 or when they satisfy square-class, norm-group, or Hilbert-symbol conditions for codimensions 0-2.