# Tagged Questions

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### What do the elements of the field $\mathbb{Z}_2[x]/(x^4+x+1)$ look like? What is its order?

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 $F = \mathbb{Z}_2[x]/(x^4+x+1)$. ...
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### The inverse of $2x^2+2$ in $\mathbb{Z}_3[x]/( x^3+2x^2+2)$

What is the independent coefficient in the inverse of $2x^2+2$ in $\mathbb{Z}_3[x]/(x^3+2x^2+2)$ ? I have been calculating some combinations, but I don't know how I can calculate the inverse.
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### Inclusion of Fields whose order is a prime power

Blue was correct, I need to fix my understanding of this: Finite fields have cardinality of a prime order because they have a prime subfield that has finite characteristic. I do not completely ...
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### Galois Group of $x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$

Question is to find Galois group of $f(x)=x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$ What i have done so far is : I could see that $f(x)$ is Irreducible and ...
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### $2^{p}\equiv1$ mod $2p+1$ for certain $p$

Let $p$ be a prime number such that $p\equiv3$ mod $4$ and $2p+1$ is also a prime number. It is well known that $2^{p}\equiv1$ mod $2p+1$, but I haven't been able to prove it.
### Zeros of $317x^{2}-151xy+40y^{2}$ over $\mathbb{F}_{31}$
Let $K:=\mathbb{F}_{31}$ and $f(x,y):=317x^{2}-151xy+40y^{2}$. I have to find out if there exists any point $(a,b)\in K^{2}$ such that $f(a,b)=0$ and $a\neq0$ or $b\neq0$.