Math 480, Spring 2014, Assignment 2

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

1. Theorem relating ideals of $\mathsf{k}[V]$ to ideals of $\mathsf{k}[x_1,\dots,x_n]$.
2. Theorem on finite generation of ideals in $\mathsf{k}[V]$.
3. Theorem identifying a basis (in the sense of linear algebra) for $\mathsf{k}[V]$.
4. Theorem relating the dimension of $\mathsf{k}[V]$ (in the sense of linear algebra) to the cardinality of $V$.

Describe the following algorithms:

1. Algorithm to compute sums and products in $\mathsf{k}[V]$ (with results expressed in terms of the basis identified above).

Solve the following problems:

1. Section 5.2, problem 1.
2. Section 5.3, problem 5.

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