# Questions tagged [finite-fields]

A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.

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### Least prime in Artin's primitive root conjecture

Let $a$ be an integer which is neither a square nor $-1$. Artin's conjecture states that there are infinitely many primes $p$ for which $a$ is a primitive root modulo $p$. My question is whether there ...
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### On a matrix problem in the field $\mathbb F_2$

Given $M$ a symmetric matrix in $\mathbb F_2^{n\times n}$ having $\mathsf{det}_\mathbb R(M)\neq0$ (non-singular in reals) and satisfying $PMP'=(M+J+I)$ or $P(M+J+I)P'=M$ where $P$ is a permutation ...
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### How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there?

How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there? As much as I have searched, I have not found any results that answer my question; not even for k = 1,2.
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### Dominating sets in subtournaments of the Paley tournament

For a tournament $T$, let $\mathrm{dom}(T)$ be the order of a smallest dominating set in $T$. Let $q$ be a prime power congruent to 3 mod 4 and let $T_q$ be the Paley tournament on $q$ vertices. Is ...
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### Question on rank of matrices over $\mathbb F_2$

$A$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $k\leq n-1$. $B$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $n$. $T$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $1$...
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### Schur Complement and depermuting an algorithm for determinant modulo $2$

Let $$M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$$ be a matrix in $\mathbb F_2^{n\times n}$ where $A\in\mathbb F_2$ and $D\in\mathbb F_2^{(n-1)\times(n-1)}$ are square. Through the determinant ...
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### On a $(k,l)$-monochromatic Hamming distance in $\mathbb F_2$?

A $(k,l)$-monochromatic edit on a matrix $M\in\mathbb F_2^{n\times n}$ is the operation $$M+A$$ where $A\in\mathbb F_2^{n\times n}$ is of rank $1$ and number of $1$'s in $A$ is $kl$ and there are $l$ ...
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### Invertible matrix by using polynomial in LDPC codes

I am studying about QC-LDPC codes. These codes can be represented by matrices or polynomial. For instance: Example. So, we have two polynomials: $a_1(x) = 1 +x$ and $a_2(x) = 1+x^2+x^4$. The second ...
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### On some rational points on an elliptic curve over finite field

Let $p\equiv3\pmod4$ be a prime. We consider the elliptic curve $E$ over the finite field $\mathbb{F}_p$ (in affine coordinates) defined by $$y^2=x^3+x.$$ Clearly the discriminant of $E$ is $-2^6$. ...
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### Polynomial form/Fourier transform of rational function over finite affine space

I am certainly going to make a mess of any serious algebraic terminology, so bear with me as I present my problem arising from a probability problem. Consider the space of sequences of $n$ zero-one ...
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### Do you know which is the minimal local ring that is not isomorphic to its opposite?

The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.
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### Explicit large finite fields in characteristic $2$

Every finite field of characteristic $2$ ist given by $\mathbb{F}_2[x]/P(x)$ for some irreducible polynomial $P\in \mathbb{F}_2[x]$. For small degree, a simple algorithm gives a way to find $P$. Is ...
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### Orbit counting polynomials over finite fields

Let $X$ be an affine variety defined over $\mathbb{Z}$ and let $G$ be an algebraic group defined over $\mathbb{Z}$. Let $q$ be a power of a prime number. We write $\mathbb{F}_q$ for the field with $q$ ...
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### Permutation induced by multiplication of finite field elements [closed]

Consider a finite field $\mathbb F$. Let $a \in \mathbb F$. Then multiplication by $a$ induces a permutation on the field elements. $0 \rightarrow 0$, $1 \rightarrow a$, $2 \rightarrow 2a$, etc. Is ...
### $\delta$-equidistributed polynomials over finite fields
I'm trying to show that a polynomial over finite (prime) field is "close enough" to being equidistributed over its range. A polynomial $p(\cdot)$ from $\mathbb{F}^n$ to $\mathbb{F}$ is \$\...