Fields with involution whose fixed field is ordered?Let K be a field with an involution ∗, meaning ⋅:K→K is an automorphism and (x∗)∗=x for all ξnK. Suppose further that the fixed field of ∗ is ordered (i.e., it can be given an ordering that is compatible with the field structure).

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Karsyn Wu · Accepted Answer

Step 1Try the Puiseux seriesE=⋃n≥1BC((x1n)), F=⋃n≥1BR((x1n))so that E=F+iF, a+ib→a−ibis the involution. By the axiom of choice (algebraically closed field of same cardinality) E is isomorphic to C. F is ordered ∑k≥−Kakxkn&amp;gt;0 iff the first non-zero ak is &amp;gt;0 It is not isomorphic to R nor to a subfield of R (as 1+mx has a square root in F for all m∈Z)

Fields with involution whose fixed field is ordered? Let

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