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} \in R | a, b \in 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}    .