I have this polynomials: $f = x^{4} + 3x^{3} + x^2 + 3 \in \mathbb{Z}_{5}[x]$, $g = x + 2 \in \mathbb{Z}_{5}[x]$
Does g + (f) have inversion in ring $(\mathbb{Z}_{5}[x]/(f),+,.)$ ?
I should found it or prove that it doesn't exist. I don't know how to start with this, could you give me some hint please?
I guess I should start with proving that polynomial f is irreducible in $\mathbb{Z}_{5}[x]$ but I don't know how to continue.
The division algorithm will give $$ x^4+3x^3+x^2+3=(x+2)(x^3+x^2-x+2)-1. $$ Thus $$ (x+2)(x^3+x^2-x+2)=(x^4+3x^3+x^2+3) +1. $$
EDIT: By dividing $f$ by $x+2$ you find a polynomial $p$ such that $$ (x+2)p(x)=f(x)+1. $$ But, $f(x)+1$ represents the unit in $(\mathbb{Z}/5)[x]/(f)$.