# Show that any endomorphism of $\mathbb Q[x]$ is $E_{g}$ for some $g$, and find all automorphisms of $\mathbb Q[x]$

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

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 $$F$$.

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