Maximal ordered fields of rank n
An ordered field F has a valuation v compatible with the order, and the rank of this valuation is called the rank of F. We say F is complete if it does not allow of an extension \(\hat F\) in which F is dense. The authors show that any maximal ordered field F of finite rank is complete, where by maximal it is meant one which has no proper ordered extension of the same rank. The main theorem establishes (i) the existence and uniqueness of a maximal ordered field F of rank n for any natural number n, and (ii) any ordered field K of rank n can be order-isomorphically embedded in F. This F is in fact the formal power series field in x with coefficients from R and exponents from the n-fold lexicographic product of the additive group of real numbers \({\mathbb{R}}\).
scientific article; zbMATH DE number 4019187