Real closed graded fields (Q2464640)

Let \((G,+,\leq)\) be a totally ordered abelian group. A (commutative, unitary) ring \((A,+,\cdot)\) is called \(G\)-graded if \(A=\bigoplus A_g\) with \(A_g\) \((g\in G)\) subgroups of \((A,+)\) such that \(A_g\cdot A_h\subseteq A_{g+ h}\) for any \(g,h\in G\). Hence any element \(a\in A\) can be formally written as \(a=\sum a_g\) with \(a_g= 0\) for all but finitely many \(g\in G\). Putting \(l(a)= \max\{g\in G\mid a_g\neq 0\}\) for any \(a\neq 0\) in \(A\), an ordering \(P\) on \(A\) is called homogeneous if \(a\in P\) implies \(l(a)\in P\). In order to describe such orderings on \(A\), homogeneous orderings on graded fields are studied: a \(G\)-graded field is a \(G\)-graded ring which is not the zero ring and in which any element of \(\bigcup A_g\) different from \(0\) is invertible. A graded field is called: (i) real if it has a homogeneous ordering, and (ii) real closed if it is real and has no proper algebraic extension by a real graded field. With these concepts an analogue to the theorem of Baer-Krull is proved and the chararacterization of real closed fields by Artin-Schreier is generalized to real closed graded fields. The latter are characterized in terms of the underlying group \(G\) and also in terms of its homogeneous elements of degree \(0\).