I have been trying to find an exampke fo an automorphism of $ℚ$.

I have tried with some simple equations, such as

$$a + n$$



(being $a$ an element in $ℚ$ and $n$ a constant)

None of the latter examples form an automorphism. There is clearly the identity, but that is kind of trivial.

So I was wondering, could someone provide an example of an automorphism of $ℚ$ besides the identity?

Any help/thoughts would be really appreciated.

  • 4
    $\begingroup$ Automorphism as a field? But any automorphism of a field fixes the smallest subfield containing $1$, which in this case is all of $\mathbb Q$. $\endgroup$ – lulu Sep 18 '17 at 14:46
  • 1
    $\begingroup$ See here. $\endgroup$ – wgrenard Sep 18 '17 at 15:01
  • $\begingroup$ If you are viewing $\mathbb{Q}$ as a group (with a single operation, rather than as a field with two compatible operations) then the "inversion" map is an automorphism. That is, if you are thinking of $(\mathbb{Q}, +)$ the the map $x\mapsto -x$ is an automorphism, while if you are thinking of $(\mathbb{Q}\setminus\{0\}, \ast)$ then $x\mapsto x^{-1}$ is an automorphism. $\endgroup$ – user1729 Sep 19 '17 at 10:05

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