Exercise 9

Let $p(x)=x^{2}+2$, $f(x)=2x+3$, and $g(x)=4x^{2}+3x+2$ be polynomials in $\mathbb{Z}_{5}[x]$.

1. Show that $p(x)$ is irreducible over $\mathbb{Z}_{5}$.

2. Hence construct a field, $F$, of 25 elements.

3. Let $I=\left\langle p(x)\right\rangle$ be an ideal of $F$. Simplify the following cosets.

   1. $(f(x)+I)+(g(x)+I)$

   2. $(f(x)+I)(g(x)+I)$