Elements of $\mathbb{F}_{2^2}: What Are They?

[mathmari]

Hello!

Which are the elements of $\mathbb{F}_{2^2}$ ?? (Wondering)

There are $4$ elements, right??

For each element $a$ in $\mathbb{F}_{2^2}$ it stands that $a^2=a \Rightarrow a^2 \in \mathbb{F}_{2^2}$, right??

 

[mathmari]

The elements are the following:

$$0,1,a,b$$

The product of two elements is in the field, so $$a \cdot b=0 \text{ or } 1 \text{ or } a \text{ or } b$$

It cannot be $0$ because $\mathbb{F}_{2^2}$ is an integral domain and since $a , b \neq 0$ the product cannot be $0$.

The product cannot ba $a$ or $b$, because then one of $a$ and $b$ must be $1$, and then the field would contain only $3$ elements.

Therefore, the product must be $1$, that means that $a \cdot b=1 \Rightarrow
b=a^{-1}$.

So, the elements are $$0,1,a,a^{-1}$$ right?? (Wondering)

Also, how could we show that $$\mathbb{F}_{2^2} \cong \mathbb{Z}_2(a)$$ where $a$ is of degree $2$ over $\mathbb{Z}_2$?? (Wondering)

Do we have to show that these two fields have the same elements?? (Wondering)