# Questions tagged [finite-fields]

Finite fields are fields (number systems with addition, subtraction, multiplication, and division) with only finitely many elements. They arise in abstract algebra, number theory, and cryptography. 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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### Which is the characteristic of a field K? [closed]

Calculate the characteristic of a field K knowing that $x^4 = x$ for each x $\in$ K.
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### Determining the distance of linear code using parity-check matrix

A matrix $H$ over $\Bbb F_2$ has as the first seven rows $I_7$ and the other rows are all of the vectors of $\{0,1\}^7$ with exactly three $1$'s per vector (weight-$3$ vectors). I know that this can ...
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### Finding LFSR periods

We have LFSR with corresponding linear automata $A \in F_2^{n\times n}, B \in F_2^n$. We know that this LFSR's generating function is $B^T * (A - I_n * x)^{-1} * S$, where $S$ is the state from $F_2^n$...
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### Characterization of an irreducible polynomial mod p

Let $p$ be an odd prime, and let $n=p^k$ where $k\ge 1$. Define $P_n$ to be the set of non-zero polynomials of degree at most $k-1$ and coefficients in $[0,p-1]$. Thus $|P_n|=n-1$. Now let $f$ be a ...
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### How to use the automorphism of extended field to represent the elements of subfield?

how to prove the content of the red line? $Z,Z(a)$are finite fields
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### Using quadratic extensions, show that there is only one field of order $p^2$ for an odd prime $p$.

I understand that this result has been proven before, but I am trying to prove it in another way that I can't seem to find on SE. The question goes as follows: Let $E$ be a quadratic extension of a ...
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### Monomial permutation polynomial [duplicate]

https://arxiv.org/pdf/1211.6044.pdf Theorem 1.14 (2): The monomial $x^n$ is a permutation polynomial of $F_q$ if and only if $gcd(n, q − 1) = 1$. Let's pick $n=2$ and $q=6$ so $gcd(2,5) = 1$ in this ...
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### Examples of Clifford algebras over finite fields

Yesterday I was in a discussion about solving an applied problem using clifford algebras over a finite field. While this is not (seemingly) disallowed by the definition of a clifford algebra (which ...
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### Can Fast Fourier Transform (FFT) be implemented using Clifford Algebra over GF(3)?

Background. Fast Fourier Transform (FFT) is an algorithm used to quickly calculate the discrete Fourier transform (DFT) of a sequence. It is widely used in signal processing, image analysis, and data ...
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### Normal basis on tower | composite fields

I'm trying to help someone with normal basis tower fields, specifically for isomorphic mapping of $GF(2^8)$ to $GF(((2^2)^2)^2)$. At the time I originally posted this question, I was familiar with ...
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### $\mathfrak{B}=((1,2,0)^t ,(2,1,2)^t ,(3,1,1)^t)$ is a basis of $\mathbb{R}^3$. For what prime numbers p is $\mathfrak{B}$ a basis of $\mathbb{F}^3_p$?

I'm not quite sure, but it seems to me that $\mathbb{F}^3_p$ is the finite field in three dimensions with $p$ elements inside it, where $p$ is the prime number(s) we are looking for. A finite field is ...
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### Let $V = \mathbb F_p^9$ and $W \subset V$ a dimension $5$ subspace. Find the number of subspaces $U \subset V$ with $\dim(U) = 6, \dim(W \cap U) = 3$

Let $K = \mathbb F_p$ be a finite field with $p$ elements where $p$ is a prime Let $V = K^9$ be a vector space, and let $W \subset V$ be a subspace of $V$ such that $\dim(W) = 5$. Find the number of ...
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### Enumerating monomorphisms of finite-dimensional $\mathbb{F}_2$-algebras

I want to enumerate the monomorphisms of finite-dimensional $\mathbb{F}_2$-algebras. Of course, each such monomorphism is a linear map between finite-dimensional $\mathbb{F}_2$-vector spaces, and so ...
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### Non-identically zero polynomials of degree $n$ over $\mathbb{F}_n$

Let $\mathbb{F}_n$ be a finite field, and $X \subset \mathbb{F}_n^n$ be of size $2^n - 1$. Must there exist $P$ such that $P \in \mathbb{F}_n[x_1, ..., x_n]$, $P$ is of degree $n$, $P$ is zero on $X$,...
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