Let $(A, +, \cdot)$ be a ring with 8 elements. Prove that $(A, +, \cdot)$ is a field $\Leftrightarrow \forall x \in A\backslash \left\{0\right\} \exists \:a, b \in \left\{0, 1\right\}$ so that $x^3 + ax^2 + bx + 1 = 0$.
I've tried to solve the $\Rightarrow$. This is what I did:
$A$ is a finite field $\Rightarrow$ $car(A) = p$, where $p$ is a prime number, but $p$ divides $8$, so $p = 2$. $(A*, \cdot)$ is a group with 7 elements so $ord(x) = 7$ or $ord(x) = 1$.
$ord(x) = 1$ $\Rightarrow x = 1 \Rightarrow \exists \:a=b=0$ $x \neq 1 \Rightarrow x^7 = 1 \Leftrightarrow (x^3+x^2+1)(x^3+x+1) = 0 \Rightarrow a=1, b=0$ or $a=0, b=1$.
If you could help me to solve $\Leftarrow$ I would be truly grateful.
Hint $\,\ 1\, =\, x(-b-ax-x^2)\,\Rightarrow\,x$ is invertible.