Written Assignment No. 3

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 $\mathbf{Z}$ denotes the ring of integers and $\mathbf{R}$ denotes the field of real numbers, then $$\left\{a+b\sqrt[3]{2}\in \mathbf{R}|a,b\in \mathbf{Q}\right\}$$ is a subring of $\mathbf{R}$.

2. If $F$ is a finite field with $\left|F\right|=q$ and $F\left[t\right]$ denotes the ring of polynomials with coefficients in $F$, then the number of elements in the ring $$A=F\left[t\right]/\left(t^q-t\right)F\left[t\right]$$ (of all congruence classes of polynomials modulo the polynomial $t^q-t$) is given by $$\left|A\right|=q^q\text{.}$$