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 $\operatorname{GF}(q)$.
5,610 questions
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Counting disjoint $k$-tuples of lines in $\mathbb F_q^n$
Let $k$ be a positive integer. How many $k$-tuples of disjoint lines are there in $\mathbb F_q^n$? Here two lines are disjoint if they do not share a point in $\mathbb F_q^n$.
I was wondering because ...
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what is a subspace of a projective space? [closed]
The following sentence shows up in a note "the number of $k$-dimensional subspaces of the projective space $PG(d,q)$ is $\binom{d+1}{k+1}_q$".
I am trying to understand what is $PG(d,q)$, ...
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Is there an interpretation of why vectors can be orthogonal to themselves in finite fields?
I'm learning linear algebra through Philip Klein's Coding the Matrix, and he defines the annihilator of a vector subspace $V\subset \mathbb{F}^n$ as $V^0=\{\mathbf{u}\in\mathbb{F}^n:\mathbf{u}\cdot\...
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Generators in the multiplicative group of the field $\mathbb{F}_{16}$ [duplicate]
The following are two theorems about finite field:
Theorem 1. Any finite field $\mathbb{F}_{p^n}$ is isomorphic to the quotient ring $\mathbb{F}_p[x] / (g(x))$, where $g(x)$ is an irreducible ...
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Let $q=2^k$ with $k$ even. Let $F=GF(q^3)$ and $G=\{x \in F ~|~ x^{q^2+q+1}=1\}$. If $x+x^q+x^{q^2}=1$ for $x \in G$,then $x=t^3$ for some $t\in G$.
Let $q=2^k$ with $k$ even. Let $F=GF(q^3)$ and $G=\{x \in F ~| ~x^{q^2+q+1}=1\}$. For $x\in G$, if $x+x^q+x^{q^2}=1$, then $x=t^3$ for some $t\in G$. Numerical data for k=2,4,6,8 indicate the above ...
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Basis for algebraic closure of $\mathbb{F}_2$
For any finite degree extension $\mathbb{F}_{2^t}$ of $\mathbb{F}_2$, we have at least a couple of canonical bases -- one being $\{1,\gamma,\ldots,\gamma^{t-1}\}$ where $\gamma$ is a primitive element,...
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What is the kernel of the Frobenius mapping from $\mathbb{F}_{p^n}$ to itself?? [duplicate]
The Frobenius map from $\mathbb{F}_{p^n}$ to itself is a ring automorphism of order $n$.
I am confused how to show it is injective.
For, if $\phi :\mathbb{F}_{p^n} \to \mathbb{F}_{p^n}$ is defined by $...
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Is there an interpretation of least squares in finite fields? [closed]
We know that the solution to the least squares problem $Ax \approx b$ is found by solving
$$
A^TAx = Ab
$$
This formula still makes sense over finite fields. Does it still have a sensible ...
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Classifying Lie algebras over a finite field
I've been given a problem I'm rather unfamiliar with, in particular classifying all Lie algebras of dimension 8 over $\mathbb{F}_2$. I thought to split it into the semisimple case and the case where $\...
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Can norm $N_{L/K}(X)$ of a polynomial $g(X) \in L[X]$ gives a factor of $f(X)$ in $K[X]$, when $g(X)$ is a factor of $ f(X)$ in $L[X]$?
Let us consider $L/K$ is a finite Galois extension and $G = \mathrm{Gal}(L/K)$ is the corresponding Galois group.
Assumptions:
$f(X)$ is a monic polynomial in $K[X].$
$g(X)$ is a monic factor of $f(...
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Interesting properties of matrices describing supplementary subspaces in nD vector spaces over GF(q)
This is a follow-on of this question, dealing with $V$, the $n$-dimensional vector space over the field $\text{GF(q)}$.
Let $S_{k}$ be the set of $k$-dimensional subspaces of $V$. It was asked a graph ...
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When $X^{n} - a$ Has Exactly $n$ Roots
Let $K$ be a finite field. Determine all natural numbers $n \ge 2$
with the property that, for every $a \in K$, the polynomial
$X^{n}-a \in K[X]$ has exactly $n$ roots in the field $K$.
Attempts (this ...
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Understanding a character calculation for arithmetic progressions of squares in finite fields
so the problem I'm interested in is to show that all sufficiently large finite fields contain an arithmetic progression of 9 distinct perfect squares.
A professor in my department had some previous ...
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Frobenius morphism of a Weil restriction
Let $p$ be a prime number, and let $q$ be a power of $p$. Let $\mathbf X$ be an algebraic variety over the algebraic closure $\overline{\mathbb F_q}$, which is equipped with an $\mathbb F_q$-structure ...
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Linear property of decoding algorithm in erasure codes
Recently, I learned about erasure codes from Richardson & Urbanke's Modern Coding Theory, and I want to know the necessary condition to make a decoding algorithm satisfy the linear property. To ...
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Show that the minimal polynomial of $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ is irreducible in $\mathbb{Z}[t]$ but becomes reducible modulo any prime $p$
Problem: Find the minimal polynomial of $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ and show that it is an irreducible polynomial in $\mathbb{Z}[t]$ which becomes reducible modulo any prime $p$.
I found ...
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Dimension of intersection of subspaces over $\mathbb F_2$
Let $x^1,\ldots,x^k\in \mathbb{F}_2^n$ and $S=\textsf{span}(\{x^1,\ldots,x^k\})$ where the span is over $\mathbb{F}_2$. One can assume that $k\leq n/10$. Let $S^\perp=\{y:\langle y,x\rangle=0 \text{ ...
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Can an automorphism of the base field change the number of rational points?
If we have a variety $X$ over a field $k$, $X^\sigma = X\to \text{Spec }k\xrightarrow{\sigma}\text{Spec }k$ is not isomorphic to $X$ as a $k$-scheme in general. But I don't feel like I am ...
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A problem of the finite fields extension in the case of infinite Galois extension in Bosch's Algebra
Bosch's reasoning on page 149:
Finally, let us have a look at an example in which we compute an infinite Galois group. Let $p$ be a prime number and $\overline{\mathbb{F}}$ an algebraic closure of the ...
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Elaborated example of finite field construction (characteristic 2) [closed]
I read the Wikipedia on finite field and cannot understand several explicit construction steps. I want an elaborated explanation on construction of $GF(4)$, $GF(8)$, $GF(16)$ fields, that answers on ...
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Existence of a Linear Recurrence Modulo $p$ with Period $p^2 - 1$
Let $p$ be a prime number. Prove there is an integer $c$ and an integer sequence $0 \leq a_1, a_2, a_3, \dots < p$
with period $p^2 - 1$ satisfying the recurrence
$$
a_{n+2} \equiv a_{n+1} - c \...
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Theorem 3.9 from Janusz - Algebraic Number Fields (Edition 2)
Unramified Extensions
The unramified extensions of a completion of a number field at a nonarchimedean prime are easily described and have a number of very special properties. We give just a few ...
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Compute the cardinal number of the splitting field over a finite field
Let $F$ be the splitting field of $f = X^{10}-1 \in \mathbb{F}_{3}[X]$. Determine the cardinality of $F$.
At first I tried to brute force by finding out an irreducible factor. I got to the ...
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rank of vectors where each element appears the same times
Let $F$ be the finite field of $p$ elements, where $p \ge 3$ is a prime. Call a vector $v \in F^{pl}$ balanced if each of $0,1,\dots,p-1$ appears in $v$ exactly $l$ times.
Let $S$ be the set of all ...
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Existence of Hyperplane that finds discrepancy?
Suppose I have a weight function $wt : \mathbb{F}_2^m \longrightarrow \{0,...,2^m\}$, such that $\sum_{i \in \mathbb{F}_2^m} wt(i) = 2^m$. If there exists $i\neq j \in \mathbb{F}_2^m$ such that $0 <...
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