# Tagged Questions

Finite fields are structures arising in abstract algebra. The order of a finite field is always a prime power, and for each prime power $q$ there is a single isomorphism type. It is usually denoted by $\mathbb{F}_q$ or $\operatorname{GF}(q)$.

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### Choose a basis of $\mathbb{F}_q/\mathbb{Z}_p$ to do inverse quickly.

Let $\mathbb{F}_q$ be the finite field with $q$ elements ($q=p^n$, $p$ is a prime). $\mathbb{F}_q$ can be regarded as a linear space over the field $\mathbb{Z}_p$ of dimension $n$. The question is: ...
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### How To Prove That The Rijndael Polynomial Is Irreducible?

I am learning about the AES algorithm which uses the finite field ${\mathbb{Z}_2[x]}\over{(p(x))}$, where $p(x)=x^8+x^4+x^3+x+1$. How do you prove that this polynomial is irreducible? I know that for ...
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### Sums involving binomial coefficients in a finite field

Consider the field $\mathbb{F}_q$ where $q=p^k$ for some prime $p$. I have some identities related to binomial coefficients over such a field, which I wish to prove. So, can someone tell me a source ...
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### Zech Logarithms

Let $a$ be a primitive element of $\mathbb{F}_{16}$ that satisfies the equation $a^4=1+a$. The logarithm of $1+a+a^2$ in $\mathbb{F}_{16}$ with base $a$ is the integer $i$ such that $0≤i<15$ ...
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### Non-separable, infinite field extensions of non-zero characteristic

I have been trying to find examples (and non-examples) of fields which are separable, finite and have characteristic equal to zero. Separable Example: $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ because the ...
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### Generalised Gauss sums

Let $\chi$ be a non-trivial Dirichlet character modulo an odd prime $p$ and let $f(x) \in \mathbb{Z}[x]$ be a polynomial. We define the generalised Gauss sum  G(\chi, f):=\sum_{y \in \mathbb{F}_p^*} ...
### Cardinality of a finite field is $p^{n}$ [duplicate]
Theorem: Let F be a finite field of characteristic p. Then p is a prime and $\left | F \right |=p^{n},\left [ \exists n>0 \right ] \in \mathbb{N}$ Note that the characteristic of F ...
A way of solving systems of polynomial equations over $\mathbb{R}$ or $\mathbb{C}$ is using homotopy continuation. Roughly speaking this method uses a homotopy that starts from some system of ...