Automorphisms and isomorphisms of real Henselian fields

Abstract:

Let $K$ and $L$ be ordered algebraic extensions of an ordered field $F$. Suppose $K$ and $L$ are Henselian with Archimedean real closed residue class fields. Then $K$ and $L$ are shown to be $F$-isomorphic as ordered fields if they have the same value group. Analogues to this result are proved involving orderings of higher level, unordered extensions, and, when $K$ and $L$ are maximal valued fields, transcendental extensions. As a corollary, generalized real closures at orderings of higher level are shown to be determined up to isomorphism by their value groups. The results on isomorphisms are applied to the computation of automorphism groups of $K$ and to the study of the fixed fields of groups of automorphisms of $K$. If $K$ is real closed and maximal with respect to its canonical valuation, then these fixed fields are shown to be exactly those real closed subfields of $K$ which are topologically closed in $K$. Generalizations of this fact are proved. An example is given to illustrate several aspects of the problems considered here.