Formulas for some diophantine approximation constants

Let M be a full 2-module in F a real number field of degree at least 3 with N(α) denoting the norm of a ∈ F. Given any nonzero number Ø in M we make the plausible conjecture that one can find a number β in M such that N(β) = N(Φ) and the algebraic conjugates of β (not including β) have ratios arbitrarily near any given numbers consistent with the complex algebraic conjugates of elements of F. We use the conjecture to give explicit formulas for some diophantine approximation constants. Without the conjecture our methods lead to corresponding lower bounds for these constants.