What Are the Automorphisms of Z[x]?

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The automorphisms of Z[x] include two primary mappings: the identity map, defined as ø(f(x)) = f(x), and the negation map, defined as ø(f(x)) = -f(x). Additionally, a more general automorphism can be expressed as ##\phi_u(f)(x)=f(ux)##, where u is a unit element in any commutative ring R. This discussion clarifies the nature of these automorphisms and their implications in the context of polynomial rings.

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Question: What are the automorphisms of Z[x]?

I know there are two automorphisms, one of which is the identity map, ø(f(x)) = f(x).

What is the other one? ø(f(x)) = -f(x) for all f(x) in Z[x]? Or does it have something to do with the degree or factorization of the polynomials? Please explain in detail because I don't know much about ring automorphisms. Thanks.
 
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Close: Try ##\phi## given by ##\phi(f)(x)=f(-x)##.

More generally, if ##R## is any commutative ring, and ##u\in R## is any unit element, then ##\phi_u## given by ##\phi_u(f)(x)=f(ux)## is a ring automorphism.
 

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