If a field $F$ is infinite, show that the ring homomorphism $\eta : F[x]\to C(F)$ is one-to-one.

Question

Here is the question I am trying to attempt:

I feel like there is a typo... it says let $p(x)=a_{n}x^{n}$. Shouldn't it be $p(x) = a_{n}^{n}\cdots + a_{1}x+a_{0}$? I feel like I must use the roots of $p(x)$ in my answer, but I am not sure how and where. Any hints on what I should do?

Edit: The duplicate answer did not help me because I am trying to prove injectivity...it has already been assumed that $\eta$ is a ring homomorphism.

Answer 1

Hint If ker(η)=0 then η is injective. Note that the only ideals of a field are the zero ideal and the unit ideal.

Answer 2

Your thoughts on the typo are right. It should be $p(x)=a_nx^n+\cdots+a_0$.

Now it asks you to show that $\eta$ is injective, which means that if a polynomial defines the zero function, then it is the zero polynomial.

Well, suppose $\eta(p)=0$ for a polynomial $p$. Then $p(\alpha)=0$ for all $\alpha\in F$. So $p$ has infinitely many roots if $F$ is infinite, which is impossible unless $p$ is the zero polynomial (nonzero polynomials have at most $\deg p$ roots). Thus we have $p=0$, so $\eta$ is injective.