Math 360, Fall 2014, Assignment 14

I must study politics and war that my sons may have liberty to study mathematics and philosophy.

- John Adams, letter to Abigail Adams, May 12, 1780

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

1. The ring $R[x]$ (where $R$ is a commutative ring).
2. The indeterminate $x$ (in $R[x]$, when $R$ is unital).
3. Degree of a polynomial (including the degree of the zero polynomial).
4. The field $D(x)$ (where $D$ is an integral domain).

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

1. Universal mapping property of the field of fractions (i.e. the theorem showing that the field of fractions is the "smallest" field into which a domain can be embedded).
2. Theorem on the degree of the product of two polynomials.
3. Theorem on the units of $R[x]$.
4. Theorem characterizing when $R[x]$ is an integral domain.

Solve the following problems:

1. Section 22, problems 1 and 5.
2. Find all units of $\mathbb{Z}_8[x]$.
3. Give an example of an infinite field of positive characteristic. Choose two random elements of your field and show how to add, multiply, and invert them.

--------------------End of assignment--------------------