MHB Are the Fields $\mathbb{F}_{2^2}$ and $\mathbb{Z}_2(a)$ Equal?

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mathmari
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Hey! :o

I have to show that $\mathbb{F}_{2^2}=\mathbb{Z}_2(a)$, where $a \in \mathbb{F}_{2^2}$ is of degree $2$ over $\mathbb{Z}_2$.

Could you give me some hints how I could show that?? (Wondering)
 
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Hi,

Every two finite fields $F,K$ with $|F|=|K|$ are isomorphic.
 
Explicitly $\mathbb{F}_{2^2}=\mathbb{Z}_2/\langle x^2+x+1\rangle=\{0,1,\xi,\xi +1\}$ ($\xi^2+\xi+1=0$). That is, $\left|\mathbb{F}_{2^2}\right|=4.$ On the other hand, $\left[\mathbb{Z}_2(a):\mathbb{Z}_2\right]=2$ and $\left|\mathbb{Z}_2\right|=2,$ so $\left|\mathbb{Z}_2(a)\right|=4.$
 
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