Some results on ordered fields (Q1069992)

In this paper we introduce two invariants of an ordered field and show several properties concerning those invariants. In section 3 we define ''order-bases'' of an ordered field K over its subfield L, which is analogous to ''transcendency bases'', and then we show that the order type of them is determined only by \(K\) and L. Using this fact, we estimate the cardinality of the conjugacy classes of involutions in Aut(F), the automorphism group of an algebraically closed field F of characteristic 0. As is well known, the conjugacy classes of involutions in Aut(F) correspond in one-to-one manner to the isomorphism classes of subfields of F with codimension 2. Therefore, as an easy consequence, the cardinality of isomorphism classes of subfields with codimension 2 in the complex number field is greater than the continuum. In section 4 we treat a special class of subfields of K (unifinitely closed subfields), and define an equivalence relation on them. Then the set of equivalence classes admits a total ordering, whose order type, we prove in section 5, has an intimate connection with the order type of order-bases in the case \(K\) is a real closed field.