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Suppose we consider a rigid extension field $F$, i.e., $\text{Aut}(F) = 1$ over the complex numbers $\mathbb C$. What is the minimal cardinality of $F$? In particular it should hold that in this case $|F| > |\mathbb C|$.

Moreover, if we replace $\mathbb C$ with any other algebraically closed field, what can one say in this case?

Any comment, reference, or pointer is highly appreciated.

All the best, Sebastian

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1 Answer 1

up vote 8 down vote accepted

Firstly, it may be worth linking to this related question on MO.

Pröhle proved that all fields of characteristic 0 can be embedded in a rigid field - see "Does a given subfield of characteristic zero imply any restriction to the endomorphism monoids of fields?" for his particular construction.

Later, Dugas and Göbel showed in "All infinite groups are Galois groups over any field" that for ${\mathbb C}$ it can be done in a field of cardinality the successor cardinal of $2^{\aleph_0}$ which is as good as you could expect.

In follow-up papers "Automorphism groups of fields I" and "Automorphism groups of fields II", they show that for any group $G$ and any field $K$, there exists an extension with automorphism group isomorphic to $G$ and cardinality $\aleph_0|K||G|$.

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thanks a lot. The $\aleph_0 |K| |G|$ construction does not apply here as it only holds for fields that are not algebraically closed. In fact, the Dugas-Göbel construction, as well as the older ones all fail for algebraically closed fields. In Pröhle's work there is an early remark that the construction fails in that case and that for algebraically closed fields it is not possible to maintain the size - that is what the lemma states with $G = \{1\}$. –  sebastian Apr 11 '11 at 15:15
Ah okay, I don't have access to the last of the D-G papers and the MathSciNet review unfortunately doesn't detail the construction or give the restrictions on the field, though of course, as you say, the cardinality must increase. However, they do state in the earlier papers that they obtain rigid extensions of ${\mathbb C}$, with the necessary jump in cardinality. –  dke Apr 11 '11 at 17:10
In the Dugas–Göbel 1997 paper, they prove exactly what the MR review states, with no further assumptions on the field. In particular, $\mathbb C$ does have a rigid extension of cadinality $2^\omega$. I have no idea where the claim that this is impossible comes from, but chances are it is based on a confusion with the fact that an extension of $\mathbb C$ with trivial endomorphism monoid has to have strictly larger cardinality, which is both easy to see and mentioned in Prőhle’s paper. –  Emil Jeřábek Mar 24 at 18:46

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