Graded transcendental extensions of graded fields (Q1774369)

A graded ring \(R=\bigoplus_{\gamma\in\Gamma}R_\gamma\) is a graded field if it is commutative and every nonzero homogeneous element is invertible. Throughout the paper, the grading group \(\Gamma\) is assumed to be a torsion-free abelian group. The set \(\Gamma_R\) of \(\gamma\in\Gamma\) such that \(R_\gamma\neq0\) may be a proper subgroup of \(\Gamma\). A \(\Gamma\)-graded field extension \(S/R\) is a pair of graded fields \(R\subset S\) with the same grading group \(\Gamma\). It is called gr-algebraic if every homogeneous element in \(S\) is a root of a nonzero polynomial with coefficients in \(R\). The author defines gr-transcendency bases for \(\Gamma\)-graded field extensions and shows that all these bases have the same cardinality, which is the sum of the transcendence degree of the field extension \(S_0/R_0\) and the rank of the quotient group \(\Gamma_S/\Gamma_R\). He also gives an explicit construction which yields all the pure gr-transcendental extensions \(S/R\) such that \(\Gamma_S/\Gamma_R\) is a torsion group. The case of graded fields associated to the canonical filtration of valued fields is also briefly considered.