We present an algorithm to calculate generators for the invariant field k(x)~G of a linear algebraic group G from the defining equations of G. This work was motivated by an algorithm of Derksen which allows the computation of the invariant ring of a reductive group using ideal theoretic techniques and the Reynolds operator. The method presented here does not use the Reynolds operator and hence applies to all linear algebraic groups. Like Derksen's algorithm we start with computing the ideal vanishing on all vectors ( ξ, ζ)for which ξ andζare on the same orbit. But then we establish a connection of this ideal to the ideal of syzygies the generators of the field k(x) have over the invariant field. From this ideal we can calculate the genarators of the invariant field exploiting a field-ideal-correspondence which has been applied to the decomposition of rational mappings before.
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