I am studying algebraic geometry and I would like to know some technique to identify radical ideals. For instance, how do I know if the ideal $I=(x-4,y^2-x)$ is radical?
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2There are algorithms that compute the radical, based on Groebner basis. You can find a reference to the original work in https://math.stackexchange.com/questions/146099/generators-for-the-radical-of-an-ideal, and an explanation of that in the book by David A. Cox, John Little, Donal O’Shea. – Mariano Suárez-Álvarez Sep 23 '22 at 22:23
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2You can also compute the quotient of your ring by this ideal and show it has no nilpotents. – KReiser Sep 24 '22 at 03:39
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Your ideal is $(x-4,y^2-4)=(x-4,(y-2)(y+2))$. It is contained in $(x-4,y-2)$ and $(x-4,y+2)$, both of which are prime — they are even maximal --- and equals their intersection. It follows that your ideal is radical.
Mariano Suárez-Álvarez
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Hello, really in the ideal $y^2-x$ appears instead of $y^2-4$... the same idea can work? – 3435 Sep 23 '22 at 22:40
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I understand your argument, but, how do you know that $(x-4, y-2)$ and $(x-4, y+2)$ are prime? – 3435 Sep 24 '22 at 04:08
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