Extending Automorphism of a Field
We recognize that $(\mathbb{Q},+,\times)$ is a subfield of $(\mathbb{R},+,\times)$. It is very easy to see that the automorphism of $\mathbb{Q}$ is just the identification. For a fast evidence, releases via the major actions:
 $f(1)=1 \Longrightarrow f(n)=n$ for all $n \in \mathbb{N}$.

$f(1)=1$ which claims that $f(x)=x$ for all $x \in \mathbb{Z}$.

$\displaystyle f \Bigl(\frac{p}{q}\Bigr) = \frac{p}{q}$, where $q \neq 0$.
One, after that makes use of the connection of $f$ and also the denseness of $\mathbb{Q}$ to confirm that the Automorphism of $\mathbb{R}$ is additionally unimportant.
My Question Given a subfield $K$ of $\mathbb{C}$ is it and also an automorphism of $K$ can it be included the entire of $\mathbb{C}$.
Firstly, you require a little bit even more to show that the only automorphism of $\mathbb R$ is the identification. You need to confirm that any kind of such automorphism is continual. For an evidence, allow $x$ be a favorable number ; so it is the square of some number ; so it is required to an additional favorable number under the automorphism. Given that favorable numbers most likely to favorable numbers, the automorphism is order  preserving, and also therefore the identification.
Second of all, this outcome is not real if you change $\mathbb R$ by $\mathbb C$. The automorphism team of $\mathbb C$ over $\mathbb Q$ is vast. This is due to the fact that $\mathbb C$ can be created by affixing uncountably several algebraicaly  independent transcendentals to $\mathbb Q$ and afterwards taking the algebraic closure. Any kind of automorphism of $\mathbb Q$ can be included an automorphism of $\mathbb C$, by first including the algebraic closure, and afterwards to $\mathbb C$. The very same argument would certainly benefit any kind of algebraic number area.
Extra basic subfields are gotten as adheres to : You attach some variety of transcendentals to $\mathbb Q$, and afterwards take an algebraic expansion of it. An automorphism of this area can be in a similar way included the entire of $\mathbb C$ as above.
This and also relevant inquiries concerning automorphism teams of algebraically shut areas (a subject I locate intriguing and also have actually invested time thinking of) are reviewed in Section 9.1 of
http://math.uga.edu/~pete/FieldTheory.pdf
Specifically, Theorem 77 solutions the OP's inquiry agreeably.
(These notes are still really harsh. Specifically there is not yet a bibliography. When this obtains treated, a citation to Paul Yale's paper will certainly remain in order : it was most definitely something I read when creating these notes.)
First of all many thanks a whole lot to George S. Secondly I lately found a paper entitled, which reviews something on Automorphisms of Complex Numbers.
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