Math 480, Spring 2013, Assignment 13

Carefully define the following terms, then give one example and one non-example of each:

1. Radical (of an ideal).
2. Radical ideal.

Carefully state the following theorems (you need not prove them)

1. Weak Nullstellensatz.
2. Strong Nullstellensatz.

Explain how to execute the following algorithms:

1. Radical membership algorithm.

Solve the following problems:

1. Working over \(\mathbb{R}\), compute the variety of \(\left\langle x^2+1\right\rangle\). Does this contradict the weak Nullstellensatz?
2. Working over any algebraically closed field, find generators for the radical of \(\left\langle x^2, y^3\right\rangle\). (Hint: find the ideal of the variety of this ideal, then use the strong Nullstellensatz.)

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Questions:

FWIW, I found a few more typos in the text.

On page 185, Proposition 6, \(J=\langle g_{1}, \dots, f_{r} \rangle\) should be \(J=\langle g_{1}, \dots, g_{s} \rangle\)

And on page 187, Lemma 10 (i) \(f(t)\cdot p_{1}(x),\dots,f(t)\cdot pr(x)\) should be \(f(t)\cdot p_{1}(x),\dots,f(t)\cdot p_{r}(x)\) --Matthew.Lehman (talk) 00:16, 9 May 2013 (EDT)

1. Don't both Hilbert's Nullstellensatz and Hilbert's Weak Nullstellensatz require algebraically closed fields? With regard to the first problem, the ideal appears only to give rise to an empty variety because the solutions aren't defined in the reals. This was used earlier in the course as an example of why the reals aren't algebraically closed. --Robert.Moray (talk) 08:42, 16 May 2013 (EDT)

Yes. All correct. --Steven.Jackson (talk) 10:42, 16 May 2013 (EDT)