what is the value of 5/3 in mod Z7 (modulo 7)

Question

What is $5x^4 / 3x^2$ in set of integers with modulo $7$($\Bbb Z7$). I got $2x^2$ but I don't think it is the correct one because $2x^2 * 3x^2$ gives $6x^4$ instead of $5x^4$

Answer 1

Note that in $\mathbb{Z}_{p}$, for any rational $\frac{a}{b} \in \mathbb{Q}$, such that $b \ne 0$. We have that $\frac{a}{b} \equiv ab^{-1} \pmod{p}$. In your case, $\frac{5}{3} \equiv 5 \cdot 3^{-1} \pmod{7} = 5 \cdot 5 \pmod{7} = 4 \pmod{7}$. The expression $\frac{5x^{4}}{3x^{2}} \in \mathbb{Z}_{7}[x]$ is equal to $4x^{2}$.