Dividing polynomials in $\mathbb{F}_7[x]$ I am trying to divide $4x^4+3x^3+2x^2+x+1$ by $2x^2+x+1$ in $\mathbb{F}_7[x]$.
Normally outside of $\mathbb{F}_7[x]$ I know that the answer would be $2x^2+(1/2)x-(1/4)$ with a remainder of $(3/4)x+(5/4)$. But because this is in $\mathbb{F}_7[x]$ and the coefficients must be in $\mathbb{F}_7$, I am confused on how to deal with the fractions. Can anyone help?

$\frac{1}{2}=\frac{1×4}{2×4}=\frac{4}{8}=\frac{4}{1}$ as 8 mod 7 is 1.Here you need to make denominator 1 by selecting suitable number.
$\frac{-1}{4}=\frac{6×2}{8}=\frac{6×2}{1}=\frac{5}{1}$. Since $-1=6 mod 7$. 
Similarly you can try other fraction mod 7.
