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.)

------------End of Assignment-------------

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)