Math 360, Fall 2013, Assignment 13

"And do you do Addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?"

- Lewis Caroll, Through the Looking Glass

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

1. Field of fractions (of an integral domain).
2. Polynomial function (from a commutative ring $R$ to itself)
3. Polynomial (with coefficients in $R$)
4. Evaluation homomorphism (from $R[x]$ to $R$).
5. Zero (or root) of a polynomial.

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

1. Universal mapping property of the field of fractions.
2. Universal mapping property of $R[x]$.
3. Relationship between $R[x]$ and the ring of polynomial functions on $R$.

Solve the following problems:

1. Section 21, problems 1 and 2.
2. Section 22, problems 1, 5, 7, 11, 13, and 15.

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