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What are the prime, maximal, and radical ideals in $\mathbb{C}[x]$ and $\mathbb{R}[x]$?

My gut feeling is that the prime are ideals in $\mathbb{C}$[x] are those which are generated by linear terms, so $(x-a_1,x-a_2,...,x-a_n)$ and the prime in $\mathbb{R}$[x] ideals are the irreducible ideals. However, I am unsure of how to prove that, if it is correct.

Then, would the maximal ideals also be prime? And would all ideals be radical?

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Where to begin...

If your gut feeling was right, and a prime ideal looked like, say, $(x-1,x-2)$, then $x-1-(x-2)=1$ is in that prime ideal, meaning $R$ is prime (a contradiction.) So this should be your first sign something in your guess is wrong.

Secondly, "would the maximal ideals also be prime?" Of course... maximal ideals are always prime in any ring.

Thirdly "would all ideals be radical?" Certainly not... you should notice right away that $(x^2)$ isn't radical in either ring.

Take some time to think about what these rings look like, and stop making these sorts of blind guesses :)

Both rings are principal ideal domains, so you never have to think about more than one generator for each ideal. You might consider thinking about the ring of integers (another easy to understand PID) for inspiration. The classifications of maximal, prime and radical ideals turn out to be exactly analogous.

Here's another path to try. Remember that $R/I$ is a field iff $I$ is maximal, $R/I$ is a domain iff $I$ is prime, and $R/I$ is reduced iff $I$ is radical. Use this to help test your ideas, and start with the integers if the polynomial rings seem too hard at first. If you solve this for the integers, you will see what to do for polynomial rings.

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