Orders and Order Closures for Not Necessarily Formally Real Fields Original Research Article

"Closures" and "orders" of fields which are not necessarily formally real are introduced here. The closed fields include real closed fields, algebraically closed fields, and the p-adically closed fields of arbitrary p-rank. The theorem of Artin and Schreier on the bijective correspondence between orderings and isomorphism classes of real closures is generalized. An isomorphism theorem for Henselian extensions is proved which generalizes theorems of Becker and of Prestel and Roquette for generalized real closed fields and Henselian p-adic fields, respectively. A basic tool is the theory of Henselizations of fields with respect to "extended" absolute values, i.e., ones which can take the value ∞.