A ring with $8$ elements

Question

Let $(A,+,\cdot)$ be a unitary ring with 8 elements. Prove that :

$a)8=0$ and $k\neq 0$, for any odd $k$.

b)If $\exists a\in A$ such that $a^3+a+1=0$, then $a\neq 0$, $a\neq 1$, $2=0$, $a^7=1$ and $A$ is a field.

a) is pretty straightforward, it is obvious that $8=0$ from Lagrange's theorem in the additive group $(A,+)$, and then if we had some odd $k$ such that $k=0$ then we would have that $1=0$, which is a contradiction to $|A|=8$.

For b) the most difficult part is showing that $\operatorname{char}A=2$.
The fact that $a\neq 0$ and $a\neq 1$ is really trivial.
Proving that $a^7=1$ is easy as well if we know that $2=0$. From here we easily get that $A$ is a field and we are done.

The only progress I made towards showing that $2=0$ was that $a$ is invertible(I guess it might help) because we have that $a(-a^2-1)=(-a^2-1)a=1$.
Obviously, we also have that $\operatorname{char}A\in \{2,4,8\}$, but I didn't get any further.

Answer 1

Suppose that $a = n \cdot 1$ for some $n \in \mathbb{Z}$. Then $\left( n^{3} +n +1 \right) \cdot 1 = 0$, and hence $n^{3} +n +1$ is even since it is a multiple of $\text{car}(A) \in \lbrace 2, 4, 8 \rbrace$. As this is not possible, $a \notin \mathbb{Z} \cdot 1$. In particular, $\mathbb{Z} \cdot 1 \neq A$, and so $\text{car}(A) \in \lbrace 2, 4 \rbrace$.

Now, assume that $\text{car}(A) = 4$. Then $A = \lbrace 0, 1, 2 \cdot 1, 3 \cdot 1, a, a +1, a +2 \cdot 1, a +3 \cdot 1 \rbrace$ as these elements are pairwise distinct. It follows that $2 \cdot a \in \mathbb{Z} \cdot 1$ since $a \notin \mathbb{Z} \cdot 1$, and hence $2 \cdot a \in \lbrace 0, 2 \cdot 1 \rbrace$ because $4. a = 0$. Therefore, we have either $2 \cdot a = 0$ or $2 \cdot (a +1) = 0$. This is impossible since both $a$ and $a +1 = -a^{3}$ are invertible and $\text{car}(A) = 4$.

Thus, we have proved that $\text{car}(A) = 2$.

Answer 2

This is by no means a complete solution, but I just wanted to share some ideas. I will use some of the ideas in the comments. I will assume that $char(A) \in \{ 2 , 4 , 8 \}$. Suppose that $char(A) = 8$. Then $R = \{0,1,2,3,4,5,6,7\}$ is a set of $8$ distinct elements. However, the image of the polynomial $f(x) = x^{3} + x + 1$ under $R$, where $R$ has characteristic $8$, is $\{1,3,5,7\}$. Hence, condition $(b)$ is not satisfied, so we may assume $char(A) \ne 8$. Suppose that $char(A) = 4$ and $2 \ne 0$. By the given equation, we have that $1 = -a^{3} - a$. Then $$ 4a = 0 \implies 2a +2a = 0\implies 2a(1-a^{2}) - 2 = 0 \implies 2a(1-a^{2}) = 2 $$ Here is where I am not quite sure, but I was thinking $2a(1-a^{2}) = 2 \implies a(1-a^{2}) = a - a^{3} = 1 = -a^{3} -a \implies 2a = 0 \implies 2 = 0$, because $a$ is unit. So if we can justify that $2a(1-a^{2}) = 2 \implies a(1-a^{2}) = 1$, then we are done.