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Having a hard time thinking of how to tackle this. It's part 1 of a much longer assignment. Hints before outright solutions would be much appreciated.

m. lekk
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    Compute the quotient. Both ideals are even prime. – MooS Mar 09 '18 at 20:43
  • I know intuitively that they're prime, and prime => radical, but I don't know how to show they're prime. – m. lekk Mar 09 '18 at 20:48
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    As I said: Compute the quotient. – MooS Mar 09 '18 at 21:06
  • If you know the Nullstellensatz, you could consider the fact that an ideal $I$ is prime if and only if the corresponding algebraic subset $Z(I)$ is irreducible. Assuming $k$ is a field, you could also try the linked question https://math.stackexchange.com/questions/501275/is-the-ideal-generated-by-an-irreducible-polynomial-prime/816406. – An Coileanach Mar 09 '18 at 21:13
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    @AnCoileanach That does not help at all, since the irreducibility of $Z(I)$ does only show that $\sqrt I$ is prime. And we want to show that $\sqrt I =I$. – MooS Mar 09 '18 at 21:17
  • @MooS, sorry you're totally right, my bad. I can delete my comment if you think it will only muddy the waters. – An Coileanach Mar 09 '18 at 21:21

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Here's some start-help on how to compute the quotient of the second ideal.

First observe that one of the generators is linear. This always means that we can eliminate one of the variables:

$$ k[x,y,z]/(x-z^2,y+z) \stackrel{y \mapsto -z}{\simeq} k[x,z]/(x-z^2) $$

Here we eliminated $y$, since in the quotient we have that $y \equiv -z$.

Do you see how you an also eliminate $x$?

Fredrik Meyer
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