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:

  1. $f(1)=1 \Longrightarrow f(n)=n$ for all $n \in \mathbb{N}$.
  2. $f(-1)=-1$ which claims that $f(x)=x$ for all $x \in \mathbb{Z}$.

  3. $\displaystyle f \Bigl(\frac{p}{q}\Bigr) = \frac{p}{q}$, 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 $\mathbb{C}$.

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2019-05-09 11:25:28
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Answers: 3

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.

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2019-05-10 08:30:24
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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.)

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2019-05-10 07:44:21
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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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2019-05-10 06:19:19
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