Structure and transcendence degree of \(\eta_{\alpha +1}\)-fields and ultrapowers of fields (Q754918)

The principal results concerns \(\eta_\alpha\)-fields (Hausdorff definition) and \(\zeta_0\)-fields (a \(\zeta_0\)-set is a well ordered set such that for every subset \(B\) there is a cofinal subset \(B_0\) of \(B\) with \(\text{card}\, B_0\leq N)\). An ultrapower of a field \(K\) is a field \(K^I/\mathfrak M\) where \(\mathfrak M\) is a maximal ideal of \(K^I\). Then the real \(\eta_1\)-fields are the ultrapowers \(\mathbb R^{\mathbb N}/\mathfrak M\); if \(\text{card}(K)\) satisfies a certain inequality there are only two isomorphic ultrapowers of \(K\), \(K^{\mathbb N}/\mathfrak M\). Let \(H\) and \(L\) be \(\eta_{\alpha +1}\)-fields extensions of a \(\zeta_0\)-field \(K\) whose transcendence degree is \(\leq\aleph_{\alpha +1}\). Then \(H\) and \(L\) are \(K\)-isomorphic. On \(K\) the author defines a class of fixed maximal ideals. If \(\mathfrak M\) is not fixed, the transcendence degree (over \(K\)) of \(K^{\mathbb N}/\mathfrak M\) is \((\text{card}\,K)^\aleph\). The last part of the article is devoted to a construction process of the \(\eta_\alpha\)-fields.