Background: I'm looking at old exams in abstract algebra. The factor ring described was described in one question and I'd like to understand it better.
Question: Let $F = \mathbb{Z}_2[x]/(x^4+x+1)$. As the polynomial $x^4+x+1$ is irreducible over $\mathbb{Z}_2$, we know that $F$ is a field. But what does it look like? By that I am asking if there exists some isomorphism from $F$ into a well-known field (or where it is straightforward to represent the elements) and about the order of $F$.
In addition: is there something we can in general say about the order of fields of the type $\mathbb{Z}_2[x]/p(x)$ (with $p(x)$ being irreducible in $\mathbb{Z}_2[x]$)?
Answer
The elements of $F$ are $\{ f(x) + (x^4 + x + 1) \mid f(x) \in \mathbb{Z}_2[x], \deg f < 4 \}$. There are $2^4$ of them. Any field of order $2^4$ is isomorphic to $F$.
In general, if $p(x) \in \mathbb{Z}_2[x]$ is irreducible of degree $k$, then $\mathbb{Z}_2[x]/(p(x))$ is a field of order $2^k$.
There is a notation that makes this field more convenient to work with. Let $\alpha = x + (x^4 + x + 1) \in F$. Then for $f(x) \in \mathbb{Z}_2[x]$, $f(\alpha) = f(x) + (x^4 + x + 1)$. So, for example, we can write the element $x^2 + 1 + (x^4 + x + 1)$ as $\alpha^2 + 1$. In this notation,
$$F = \{ f(\alpha) \mid f(x) \in \mathbb{Z}_2[x], \deg f < 4 \}.$$
An isomorphic field is the nimber field of nimbers less than 16. The representation of the elements is simpler, but I'm finding nim-multiplication to be harder than polynomial multiplication (maybe there's a trick to it that I don't know).
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