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A real subfield $R\subseteq F$ of any number field has (finitely many) maximal real intermediate fields $R\subseteq R'\subseteq F$. Can I call such an $R'$ a real closure of $R$ relative to $F$?

Usually a closure operation has some relevant uniqueness, which this has not. For my purposes the interesting case is where $F$ is normal over $\mathbb{Q}$, and then the maximal intermediate real field is unique up to isomorphism over $\mathbb{Q}$, but generally not over $R$. This suggests closure is not a good term though it is handy for my purposes. Is there some more established term?

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  • $\begingroup$ When you say real, do you mean it has at least one real embedding, or all of its embeddings are real? Or are you fixing ahead of time an embedding of $F$ into $\mathbb{C}$? $\endgroup$
    – Kevin Ventullo
    Commented Feb 1, 2014 at 15:33
  • $\begingroup$ I mean $-1$ is not a sum of squares in it, or if you prefer it has at least one real embedding. And I do not assume a chosen order or embedding in $\mathbb{C}$ $\endgroup$
    – Colin McLarty
    Commented Feb 1, 2014 at 15:40

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