Okay. I showed that $(x)$ is a maximal ideal in the polynomial ring $F[x]$, where $F$ is some field. Now I have been asked to find another maximal ideal in $F[x]$. I tried showing that $(x+1)$ is a maximal ideal, but I had to luck. I could use a hint. I don't know very much about polynomials at this point (e.g., the degree of a polynomial hasn't even been defined yet; also, I don't know that $F[x]$ is a PID). I do know, e.g., that if $F[x]/I$ is a division ring, where $I$ is an ideal, then $I$ is a maximal ideal. I tried this for $I=(x+1)$, but I had no luck.
[Math] Maximal Ideals in Polynomial Ring Over a Field
abstract-algebrafield-theorymaximal-and-prime-idealsring-theory
Best Answer
Hint: Instead of describing another maximal ideal explicitly using generators, just try to describe it as a kernel of some surjective homomorphism to a field. The ideal $(x)$ is the kernel of surjective homomorphism $\varphi:F[x]\to F$ given by $\varphi(f)=f(0)$. Can you think of any other homomorphisms $F[x]\to F$ defined similarly, which you could take the kernel of?
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