Written Assignment No. 3

due November 11, 2005

General Directions: Written assignments should be submitted typeset. What you submit must represent your own work.

Assigned Exercises

Read these directions carefully: For each of the following statements either provide a proof that the statement is true or label the statement as false and provide justification.

1. If Z denotes the ring of integers and R denotes the field of real numbers, then

\{a + b\sqrt[3]{2} ∈ R | a, b ∈ Q\}

is a subring of R.

2. If F is a finite field with |F | = q and F [t] denotes the ring of polynomials with coefficients in F, then the number of elements in the ring

A = F [t]/(t^{q} − t)F [t]

(of all congruence classes of polynomials modulo the polynomial t^{q} − t) is given by

|A| = q^{q} .