My question today is: Is polynomial $1+x+x^2$ irreducible in $\mathbb{Z}_5[x]?$ Let $[1+x+x^2]$ be the ideal of $\mathbb{Z}_5[x]$ generated by $1+x+x^2$. Is the quotient ring $\mathbb{Z}_5[x]/[1+x+x^2]$ a field? How many elements does the quotient ring $\mathbb{Z}_5[x]/[1+x+x^2]$ have?
I've answered most of it but I just have to answer this: How many elements does the quotient ring $\mathbb{Z}_5[x]/[1+x+x^2]$ have?
I know the polynomial is irreducible, and I know it is a field. But how can I answer the last part? I see on the internet that I have to use the division algorithm, but what should I divide? I hope some of you can help me with the last part.
The order of a finite field in general will be $p^n$ where $p$ is the characteristic of the field and n is the degree of the polynomial(or equivalently the dimension of the field over $(1_F)$. You could also go the long route and construct the field explicitly, but it does not appear necessary in this case.
Hint: The characteristic of the field will be equal to the characteristic of $\mathbb{Z}_p$