Does $\operatorname{Con}\sf(ZF)$ imply $\operatorname{Con}\sf(ZF + \operatorname{Aut}{\bf C = Z/\mathrm 2Z})$?

Question

How many field automorphisms does $\mathbf{C}$ have? If you assume the axiom of choice, there are tons of them -- $2^{2^{\aleph_0}}$, I believe. And what if you don't -- how essential is the axiom of choice to constructing "wild" automorphisms of $\mathbf{C}$? Specifically, if you assume that $\mathsf{ZF}$ admits a model, does that imply that $\mathsf{ZF}$ admits a model where $\mathbf{C}$ has no wild automorphisms: $\operatorname{Aut}(\mathbf{C})=\mathbf{Z}/2\mathbf{Z}$?

I suppose if that's true, then the next logical question is to construct models of $\mathsf{ZF}$ where $\operatorname{Aut}(\mathbf{C})$ has cardinality strictly between $2$ and $2^{2^{\aleph_0}}$--pretty disturbing if you ask me. Which finite groups can you hit?

Answer 1

The use of inaccessible cardinals is not necessary here, the Baire property works just as well as Lebesgue measure. Shelah (Can you take Solovay's inaccessible away, Isr. J. Math. 48, 1984, 1-47) shows that $\mathsf{ZF}$ + $\mathsf{DC}$ + "every subset of $\mathbf{R}$ has the Baire property" is relatively consistent with $\mathsf{ZF}$. (This is also the paper where Shelah also shows that the inaccessible cardinal is necessary for Solovay's result.)

The connection is an old theorem of Banach and Pettis which says that any Baire measurable homomorphism between Polish groups is automatically continuous. This result is provable in $\mathsf{ZF}$ + $\mathsf{DC}$. Since $\mathbf{C}$ is a Polish group under addition, it follows that every additive endomorphism of $\mathbf{C}$ is continuous in Shelah's model. Since the continuous additive endomorphisms of $\mathbf{C}$ are precisely the $\mathbf{R}$-vector space endomorphisms, it follows that the only field automorphisms of $\mathbf{C}$ in Shelah's model are the identity and conjugation.

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As pointed out by Pete Clark in the comments, the Artin-Schreier Theorem goes through using only the Boolean Prime Ideal Theorem ($\mathsf{PIT}$), which is significantly weaker than full $\mathsf{AC}$. This shows that $\mathsf{AC}$ is not completely necessary to show that there is a unique conjugacy class of elements of order $2$ in $\mathrm{Aut}(\mathbf{C})$ and that these correspond precisely to the finite subgroups of $\mathrm{Aut}(\mathbf{C})$.

Looking at Pete Clark's Field Theory Notes, specifically at Steps 4 and 5 of his proof of the Grand Artin-Schreier Theorem on pages 62-63, I think that it is a theorem of $\mathsf{ZF}$ that the only possible order for a nontrivial finite subgroup of $\mathrm{Aut}(\mathbf{C})$ is $2$.

Answer 2

Pete's answer is informative. But there is a subtle point that actually turns the answer somewhat upside down. It turns out that the answer is related to large cardinals! [Edit: François's answer shows how to avoid the inaccessible cardinal.]

Pete mentions the Kestelman article Automorphisms of the field of complex numbers, which explains

Every function which defines a non-trivial automorphism of the complex numbers transforms every bounded set (in the Argand plane) into a set of Lebesgue measure zero or else into a non-measurable set.

By considering larger and larger bounded sets, this means that the existence of a nontrivial automorphism implies the existence of a nonmeasurable set. I believe that this part of the Kestelman article does not use AC, although I suppose that one must have Dependent Choices (DC) to have a decent theory of Lebesgue measure.

Pete mentions that there are known to be models of ZF in which every set of reals is measurable. These models, however, as Gerald mentions in his comments, are constructed from a ground model of ZFC having an inaccessible cardinal (Solovay's model). Shelah has proved that this large cardinal hypothesis cannot be omitted. Thus, the consistency of ZF + DC + "Every set is Lebesgue measurable" is equivalent to the theory ZFC + "there is an inaccessible cardinal". One way to explain what this means is that we should be exactly as confident in the consistency of inaccessible cardinals as we are that there is no analogue of the Vitali construction of a non-Lebesgue measurable set not using AC.

Since the Kestelman result shows that the existence of a nontrivial automorphism of C (in the presence of DC) implies the existence of a nonmeasurable set, this establishes:

• Con(ZFC + "there is an inaccessible cardinal") implies Con(ZF + DC + "there is no nontrivial automorphism of C").

This is the actual result that Pete's argument provides. The hypothesis here is strictly stronger than Con(ZF), if ZF is consistent. [Edit: François shows that by using the Baire property instead of measure, one avoids the need for inaccessible cardinals, so he has the optimal argument.]

Having DC in the conclusion seems what should be desired, when considering functions on R and C, since even to know that the epsilon–delta and convergent sequence characterizations of continuity are equivalent uses DC.

I'm not sure what happens if one drops DC in the conclusion. For example, it is known to be consistent with ZF that the reals are a countable union of countable sets, and this model does not have DC or even countable choice. Perhaps this is a good candidate model?

Finally, the question about realizing other groups is extremely interesting.

Answer 3

First things first: assuming AC, it is indeed true that for any algebraically closed field $F$, $\# \operatorname{Aut}(F) = 2^{\# F}$. The main idea for this is that we can choose a transcendence basis and then every permutation of the transcendence basis extends to an automorphism of $F$. See e.g. Theorem 80 on p. 49 of Field Theory for more details.

Second, yes, it is consistent with ZF that $\operatorname{Aut}(\mathbb{C})$ is just the identity and complex conjugation, at least if you believe in inaccesible cardinals. There are a lot of results of the form "A field automorphism of $\mathbb{C}$ is continuous (i.e., is the identity or complex conjugation) if…." One of these sufficient conditions is measurability, e.g.

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Kestelman, H. Automorphisms of the field of complex numbers. Proc. London Math. Soc. (2) 53, (1951). 1–12.

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And it is well-known I believe I have heard that there are models of ZF in which every subset of $\mathbb{C}$ is measurable. [Addendum: As Prof. Edgar mentions in his comment, there is the Solovay model, whose construction relies on the existence of an inaccessible cardinal. So I'm not sure whether it is known unconditionally whether "All subsets of $\mathbb{C}$ are Lebesgue measurable" is consistent with ZF. But it seems that this is believed to be true, at least.]

As for your third question — in conventional mathematics we have the Artin–Schreier theorem, which implies that for any algebraically closed field $F$, $\operatorname{Aut}(F)$ has no finite subgroups of order greater than $2$. (See e.g. loc. cit., Theorem 98 on p. 61.) But the proof of this uses AC. Without AC, I certainly don't know. I suspect you'll need an actual set theorist (such people exist on MO!) for that.

Addendum: As established in a previous MO question — How much choice is needed to show that formally real fields can be ordered? — the orderability of all formally real fields is equivalent to the Boolean Prime Ideal Theorem. It follows that (as François G. Dorais suggested) BPIT implies the Grand Artin-Schreier Theorem. Perhaps this could be helpful in answering Jared's last question (though I don't immediately see how).