Mrthdrb

I am presented with $f(x) = 2x^3 + 3x^2 + 1$ $in mathbbZ_5[x]$ and need to explain why $F = fracmathbbZ_5[x]f(x)$ 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 $mathbbZ_5[x]$, then the factor ring $fracmathbbZ_5[x]f(x)$ 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

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 $Bbb Z_p[x]/langle frangle$ has $p^n$ elements and is a vector space over $Bbb Z_p$ of dimension $n$.

Hint: Use the division algorithm to find canonical representatives for each residue class $g(x) + (f(x))$.

Outline:

You know $Bbb Z_5[x]/I$ where $I=langle f(x) rangle $ is a field using the mentioned result. Now, the task is to find $|Bbb Z_5[x]/I |$. Here $Bbb Z_5[x]/I$ can be considered as an entension of $Bbb Z_5$. Also $$dim_Bbb Z_5 (Bbb Z_5[x]/I)=3=deg f$$

Call $v_1,v_2,v_3$ the basis of $Bbb Z_5[x]/I$ over $Bbb Z_5$

Now arbitrarily pick $f(x)+I in Bbb Z_5[x]/I$. Then $$f(x)+I=a_1v_1+a_2v_2+a_3v_3;;;a_i in Bbb Z_5$$

There are $5 times 5 times 5$ choices to choose the coefficients, so $$textnumber of such $f(x)+I$ =5^3=|Bbb Z_5[x]/I|$$

Note that two polynomials $g(x)$ and $h(x)$, both of degree 2 or less, will represent different elements in $F$. This is due to the fact that their difference
$g(x)-h(x)$ being of degree less than 3, cannot be a multiple of the cubic polynomials $f(x)=2x^3+3x^2+1$, and so belong to different cosets.

Now a polynomial $h(x)$ of degree 3 or more can be divided by $f(x)$ to get quotient $q(x)$ and remainder $r(x)$
such that $h(x)=q(x)f(x) + r(x)$

Clearly $r(x)$ and $h(x)$ will represent the same coset. This shows that the field $F$ in your question simply consists of polynomials of degree less than $3$. This is easy to count as each coefficient has only a finite number of choices.