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What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,...]$?

(this is the ring of polynomials over the reals with countably infinite many indeteminates).

My attempt: I think taking the principal ideal generated by an irreducible polynomial gives a prime ideal.

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    @QiaochuYuan: Oh, I see. I should think about why it's prime and why there are no other primes. Thanks for the help.2012-06-10

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