Find prime number $p$ and polynomial $f$, so that the ring $F_p[x]/(f)$:
1) contains non-zero element $w$: $w^n=0$ for some $n$
2) doesn't satisfy 1), but is a field
3) is a field which contains $8$ elements,
4) $9$ elements
I know definitions of ring and field, but I'm just starting solving examples, so I'm asking for any help how to start it.
On
(1) For example, for any prime $\;p\;$:
$$\Bbb F_p[x]/\langle x^2\rangle\;,\;\;\text{with}\;\;w:=x+\langle x^2\rangle$$
(2) For example with $\;f(x)=x\;$
For (3) you need precisely $\;p=2\;$ , and for (4) you have to work with $\;p=3\;$ , and in both cases you need an irreducible quadratic over the correspondent $\;\Bbb F_p\;$ . Try, for example $\;x^3+2x+1\;$ for (4), and try to do by yourself (3)
You need to think in terms of polynomials which are irreducible (or reducible depending on what you want) in a given field $\mathbb{F}_p$.
For example, Consider $\mathbb{F}_2[x]/\langle x^2+1 \rangle$. It has $0,1,x,x+1$ as its elements (think why?). Now consider $(x+1)^2 \equiv x^2+2x+1 \equiv x^2+1 \equiv 0$ in this ring. Thus it has an element $w=x+1$ such that $w^2=0$.
Now try with a different polynomial say $x^2+x+1$ and see what kind of ring structure you will get.