I am presented with $f(x) = 2x^3 + 3x^2 + 1$ $\in \mathbb{Z_5}[x]$ and need to explain why $F = \frac{\mathbb{Z_5}[x]}{f(x)}$ is a field and also find how many elements are in F.
So far I have shown that $f(x)$ is an irreducible polynomial and I also know that,
If $f(x)$ is an irreducible polynomial in $\mathbb{Z_5}[x]$, then the factor ring $\frac{\mathbb{Z_5}[x]}{f(x)}$ is also a field.
Basically I am not sure how to properly find the factor ring F and also determine how many elements are in it.
Thanks in advance
Best Answer
Well, in general, if a field $L$ is an extension field of some field $K$, then $L$ is also a $K$-vector space.
If $f$ is an irreducible polynomial of degree $n$ over, say, ${\Bbb Z}_p$, then the quotient field ${\Bbb Z}_p[x]/\langle f\rangle$ has $p^n$ elements and is a vector space over ${\Bbb Z}_p$ of dimension $n$.