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I have a homework problem to prove that if $F$ is a field, $p(x)$ is in $F[x]$, and $\langle p(x) \rangle$ is a maximal ideal in $F[x]$, then $p(x)$ is irreducible over $F$. My book provides an answer, and going through it I understand it, but I thought of a possible alternative proof and I wanted to know if anyone could check if this is correct?

My proof is that since $\langle p(x) \rangle$ is maximal, $F[x]/\langle p(x) \rangle$ is a field, and thus an integral domain. That implies that $p(x)$ is prime, and hence irreducible.

It seems a lot simpler than what the book gave (shown in pic), which made me suspicious that it might not be right. Thanks to anyone who helps!

enter image description here

Accelerator
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cowboylikeme
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    I think your proof is correct. – PinkRabbit Apr 04 '23 at 05:21
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    I made your pseudo-math into LaTeX form. Can you check if it is what you intended to write? – Accelerator Apr 04 '23 at 06:20
  • Yes, it's a common proof, e.g. see the linked dupe. Please note that for a solution-verification question to be on topic you must specify precisely which step in the proof you question, and why so. This site is not meant to be used as a proof checking machine. It's best to delete this quesiton since we already have too many posts on this FAQ. – Bill Dubuque Apr 04 '23 at 06:40

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