Math 480, Spring 2013, Assignment 1

The beginner ... should not be discouraged if ... he finds that he doesn not have the prerequisites for reading the prerequisites.

- P. Halmos

Solve the following problems:

1. Name two familiar arithmetic structures that are fields, and two familiar arithmetic structures that are not.

1. Find an integer \(n\) for which the structure \(\mathbb{Z}_n\) is not a field. Write the multiplication table for this structure, and use the table to explain why, for this choice of \(n\), \(\mathbb{Z}_n\) is not a field. Then choose another \(n\) for which \(\mathbb{Z}_n\) is a field, and use the resulting multiplication table to verify this.

1. Find a non-zero polynomial with coefficients in \(\mathbb{Z}_7\) whose associated function vanishes identically.

1. What is the degree of the polynomial \(x^3 + x^2yz\)?

1. Give an example of a field which is not algebraically closed, and prove that it is not.

Carefully state the following theorems:

1. Theorem on vanishing polynomial functions over an infinite field (Proposition 5 in the text).