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Following the question What is the size of the smallest rigid extension field of the complex numbers?, where it was noted that the least cardinality of a rigid field containing $\mathbb{C}$ is $(2^{\aleph_0})^+$ I have the following question:

Is the rigid field of that cardinality (I.e. $(2^{\aleph_0})^+$) containing $\mathbb{C}$ unique (as a field)?

Or, Is there a canonical one?

PS: By "rigid field" I mean a field with non nontrivial field automorphisms; references are provided in the answer to the question I mentioned.

Edit: The constraint on the cardinality mentioned above may not be true. It was stated in the comments that the least cardinality of a rigid field containing $\mathbb{C}$ is rather $2^\omega$.

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  • $\begingroup$ You should have reminded that "rigid field" means "with no nontrivial field automorphism", and have provided a link: (mathoverflow.net/questions/61058/…). Btw I'm puzzled by this question because by rigid extension of $K$ they mean "with trivial automorphism group as $K$-algebra", which then includes $K$. So it would be useful to clarify, in particular I'd like a precise statement and reference for the fact that there is no rigid extension (in which sense?) of $\mathbb{C}$ of cardinal $2^{\aleph_0}$. $\endgroup$
    – YCor
    Commented Mar 22, 2015 at 11:39
  • $\begingroup$ I edited to add the link. Note that "unique" still has two possible meanings (as a field, or as a $\mathbf{C}$-field). Moreover the question you link at defines "rigid" with another meaning than yours (considering only automorphisms as $\mathbf{C}$-algebra). $\endgroup$
    – YCor
    Commented Mar 22, 2015 at 11:45
  • $\begingroup$ Thank you; Maybe "unique" should mean as a $\mathbb{C}$-field. $\endgroup$
    – user38200
    Commented Mar 22, 2015 at 11:49
  • $\begingroup$ And you still define "rigid" as an abstract field? Besides I still can't find in the linked question any reference for the result that an extension $K$ of $\mathbf{C}$ of cardinal $2^{\aleph_0}$ has nontrivial automorphisms (of field? of $\mathbf{C}$-field assuming $K\neq\mathbf{C}$)? $\endgroup$
    – YCor
    Commented Mar 22, 2015 at 11:51
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    $\begingroup$ I finally managed to get a copy of Dugas&Göbel (1997), Automorphism groups of fields II. They prove exactly what is stated in the MR review: for every group $G$ and field $F$, there is an extension $K$ of $F$ whose automorphism group is $G$, such that $|K|=\aleph_0\lvert G\rvert\lvert F\rvert$, with no further assumptions. So, $\mathbb C$ has a rigid extension of cardinality $2^\omega$. $\endgroup$
    – Emil Jeřábek
    Commented Mar 24, 2015 at 18:40

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