I'm doing Exercise 3 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.

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For a commutative ring $R$ and $c \in R[x]$, the map $E_c: R[x] \to R[x]$ is a homomorphism such that $E_c(r) = r$ for all $r \in R$ and $E_c (x) = c$. Show that any endomorphism of $\mathbb Q[x]$ is $E_{g}$ for some $g$. Find all automorphisms of $\mathbb Q[x]$.

Could you please verify if my understanding is correct? Thank you so much for your help!

Let $\sigma$ be an endomorphism of $\mathbb{Q}[x]$. Clearly, $\sigma$ leave all elements of $\mathbb Q$ fixed. As such, $\sigma (\sum a_n x^n) = \sum a_n \sigma(x)^n$. Hence $\sigma$ is $E_{g}$ for some $g \in \mathbb{Q}[x]$. The second claim follows from an exercise appearing before.

If $F$ is a field, the group of all those automorphisms of $F[x]$ which leave all elements of $F$ fixed, consists of substitutions given by $x \mapsto a x+b, a \neq 0$ and $b$ in $F$.

  • 2
    $\begingroup$ At first glance, it certainly looks correct to me. $\endgroup$ – Lubin Aug 29 '20 at 19:18
  • $\begingroup$ Many thanks for your verification @Lubin! Nice to see you. $\endgroup$ – Akira Aug 29 '20 at 19:54

@Lubin's comment solves my problem. I post it here to remove my question from answered list.

At first glance, it certainly looks correct to me.


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