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Let the set of real numbers $\mathbb{R}$ be endowed with the natural affine structure (that means the group action is the usual addition). The question is how to prove that the only affine automorphism in affine space $\mathbb{R}$ is the identity, using the fundamental theorem of affine geometry. The affine automorphism is an affine map which is an automorphism. The fundamental theorem of affine geometry states as follows:

Let $X,X'$ be affine spaces of the same affine dimension $\ge2$. Let $f:X\to X'$ be a set-theoretical bijection which takes any three collinear points in $X$ into collinear points in $X'$. Then $f$ is semiaffine.

I have no idea know how to relate the fundamental theorem of affine geometry to the problem in question. In particular, the affine space $\mathbb{R}$ has affine dimension $1$, not $\ge2$ as stipulated in the theorem. What contradiction can I get if I assume there is a non-identity automorphism in affine space $\mathbb{R}$? I would appreciate your help greatly if you can direct me how to use this powerful fundamental theorem.

PS, for reference, the screenshot of the proposition and the fundamental theorem is copied below: enter image description here

user5280911
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  • I don’t have a clue what this is about, but why shouldn’t the reflection (multiplication by -1) be an affine automorphism? I don’t see any geometric or algebraic obstructions... – Jonas Linssen Dec 13 '20 at 11:34
  • @PrudiiArca: I added the screenshot of the text containing the proposition so my question does not look like totally a bullsh*t. – user5280911 Dec 13 '20 at 11:48
  • @Morgan Rodgers: I thought the "automorphism" in proposition 2.6.4 means affine automorphism, not field automorphism, because affine automorphism is a concept introduced in the text. Are you sure the "automorphism" in proposition 2.6.4 means field automorphism? That would be a big "Whoops ..." – user5280911 Dec 13 '20 at 12:05
  • @user5280911 I thought a little more about what this proposition had to do with the fundamental theorem and replaced my comments with what I hope is an answer to your question. – xxxxxxxxx Dec 13 '20 at 12:21

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The statements about $\mathbb{R}$ and $\mathbb{C}$ in the Proposition are about field automorphisms. The point is that

  1. It is well known that the only field automorphism of $\mathbb{R}$ is the identity (the fact that the state this is what tells you they are talking about field automorphisms, since there are many examples of affine bijections of the real number line that are not field automorphisms).
  2. We can view $\mathbb{C}$ as a 2-dimensional affine geometry over $\mathbb{R}$.
  3. We can use the fact that a "continuous automorphism" of $\mathbb{C}$ must be an affine automorphism, that is, it is both an affine map in the sense of the fundamental theorem of affine geometry, and also a field automorphism.

This lets you use the Fundamental Theorem (which is about affine maps) to classify the continuous field automorphisms of $\mathbb{C}$. Notice that there is no claim about the transformations of $\mathbb{R}$ as an affine geometry; indeed the Fundamental Theorem does not apply here because $\mathbb{R} = \mathbb{R}^{1}$ is one-dimensional.

xxxxxxxxx
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  • Thanks but I can't help thinking the author has more to say than field automorphisms. So I still think the automorphism in the proposition means affine automorphism. An intuitive argument is that what the author wants to say is that there is only one affine structure for R. We know in math, "isomorphism" means the same thing under the structure in question. So, that there is only one affine automorphism perhaps means there is only one affine structure that can be assigned to R. For example, PrudiiArca commented that reflection looks like an affine automorphism, but I think even under ... – user5280911 Dec 13 '20 at 13:02
  • this reflection map, something specific to affine space and its property remains unchanged. For example, the lines are still lines, even though they are now up-side down. The intersection relation of lines are not affected under reflection either. As a result, R and its reflection (multiplication by -1) are essentially the same thing in terms of affine structure. I think it is the idea that the proposition wants to convey, which is more than field automorphism. – user5280911 Dec 13 '20 at 13:04
  • @user5280911 No, not at all. They are definitely talking about field automorphisms in the Proposition. That is why they specifically exclude multiplication by $-1$ on the reals (it is not a field automorphism), and they also exclude many other continuous affine maps of the complex plane aside from complex conjugation (which is the only continuous affine map that is also a field automorphism). – xxxxxxxxx Dec 13 '20 at 13:10
  • Thanks for the clarification. I need some time to convince myself. – user5280911 Dec 13 '20 at 13:19
  • I think I got it. Or more accurately, field automorphism is the only way out for me to understand the proposition. Thank you for the clarification. – user5280911 Dec 13 '20 at 18:22
  • @user5280911 It was a good question to ask, I'm glad I could help clarify. – xxxxxxxxx Dec 13 '20 at 18:26