Math 361, Spring 2018, Assignment 4

Read:

1. Section 23.

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

1. $R[x_1,\dots,x_n]$ (in terms of $R[x_1,\dots,x_{n-1}]$). (As an example, give an example of a typical element.)
2. $D(x_1,\dots,x_n)$ (where $D$ is an integral domain). (As an example, give an example of a typical element.)
3. Generalized evaluation homomorphism.
4. Principal ideal.

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

1. Universal mapping property of $R[x]$.
2. Universal mapping property of $R[x_1,\dots,x_n]$.
3. Theorem concerning long division of polynomials.
4. Factor theorem.
5. Bound on the number of roots of a non-constant polynomial.
6. Containment criterion for principal ideals.
7. Equality criterion for principal ideals.

Solve the following problems:

1. Section 23, problems 1, 3, 9, 12, and 13.

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