Math 380, Spring 2018, Assignment 3

No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....

- Morris Kline, Mathematics in Western Culture

Read:

1. Section 1.4.

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

1. Ideal (in $\mathsf{k}[x_1,\dots,x_n]$).
2. $\left\langle S\right\rangle$ (the ideal generated by a set $S$ of polynomials).
3. $\left\langle f_1,\dots,f_s\right\rangle$ (the ideal generated by the finite set $\{f_1,\dots,f_s\}$).
4. $\mathbb{I}(T)$ (the ideal of the set of points $T\subseteq\mathsf{k}^n$).

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

1. Theorem relating $\mathbb{V}(S)$ to $\mathbb{V}(\left\langle S\right\rangle)$.
2. Theorem characterizing when $\left\langle f_1,\dots,f_s\right\rangle\subseteq\left\langle g_1,\dots,g_t\right\rangle$.
3. Theorem characterizing when $\left\langle f_1,\dots,f_s\right\rangle=\left\langle g_1,\dots,g_t\right\rangle$.
4. Theorem concerning the inclusion-reversing and inflationary character of the fundamental pairing $(\mathbb{V},\mathbb{I})$.

Solve the following problems:

1. Section 4, problems 1, 3, 5, 6(a), 6(b), 7, and 8. (In problem 6, the word "basis" means "generating set.")
2. Prove that, for any set $T\subseteq\mathsf{k}^n$, the set $\mathbb{I}(T)$ is always an ideal of $\mathsf{k}[x_1,\dots,x_n]$.
3. Working in $\mathbb{R}^1$, describe $\mathbb{V}(\mathbb{I}(\mathbb{Z}))$. (In topological language, this exercise shows that the "Zariski closure" of a set may be quite different from its ordinary closure. This is because the Zariski topology is extremely "coarse.")

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Questions:

This came up because of exercise 1.4.7 but is just a general question: Are we allowed to assume we're working in the smallest affine space that will fit around the polynomials concerned or does the answer boil down to the same thing whether you do your computations in k[x,y] or k[x,y,z,..,w]?