Galois extensions and \(O^*\)-fields. (English) Zbl 07673096
Summary: A field \(F\) is \(O^*\) if each partial order that makes \(F\) a partially ordered field can be extended to a total order that makes \(F\) a totally ordered field. We use the theory of infinite primes developed by Dubois and Harrison to prove the following. For a subfield \(F\) of \(\mathbb{C}\) that is finite-dimensional over \(\mathbb{Q}\), we prove that when \(F\) is Galois over \(\mathbb{Q}, F\) is an \(O^*\)-field if and only if is a subfield of \(\mathbb{R}\). We find other conditions that make \(F\) an \(O^*\)-field and provide several examples. As well for an arbitrary field of characteristic 0, we characterize the maximal partial orders that are Archimedean.
MSC:
| 06F25 | Ordered rings, algebras, modules |
Keywords:
Galois extension; infinite prime; \(O^*\)-field; normal closure; number field; Archimedean maximal partial orderReferences:
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