All Questions
Tagged with polynomials finite-fields
797
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Degree of factors of the Artin–Schreier polynomial in $\mathbb{F}_q$. [duplicate]
Consider the field $\mathbb{F}_q$, where $q$ is a power of $p$, say $q=p^n$. Let $f=x^q-x-a\in\mathbb{F}_q[x]$, with $a\in\mathbb{F}_q$.
I'm trying to determine the degree of the irreducible factors ...
- 329
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Cardinality of the zero locus of a degree 2 homogeneous polynomial on Z/2Z: avoiding Chevalley-Warning
I have never developed sufficient knowledge in algebraic geometry but I ran into an apparently easy problem, so I apologise in advance if my question sounds naive.
Suppose to have a degree $2$ ...
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Irreducible polynomials in $\mathbb F_q[T]$
Let $q$ be a power of a prime $p$. Is there an infinite set $S$ of $\mathbb N$ such that for every $l\in S$, the polynomial $T^{q^l}-T-1$ is irreducible in $\mathbb F_q[T]$.
It looks like Artin-...
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Factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$
How do I factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$?
I checked that the discriminant $D = 16 -4 = 12$ is not a square ($12^{14} = -1 \mod 29$) so this polynomial has no roots. Therefore it's ...
- 2,636
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Distinct derivations of polynomial over finite field
I am a student studying algebra and cryptography.
I wonder below question is possible.
Can I make some polynomials $f(x)$ over finite field that all derivations $f^{(k)}(x)$ are distinct when x is ...
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1
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Calculating the gcd of two polynomials in integers using a prime field
Let $f, g \in \mathbb{Z}[x]$. Let also $h \in \mathbb{Q}[x]$ be the $\gcd(f,g)$ found by the Euclidean algorithm. Now, for $p$ an odd prime, let $h^* \in \mathbb{Z}/p\mathbb{Z}[x]$ be the $\gcd(f,g)$ ...
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Find $m$ degree $q^{m-1}$ polynomials which give a bijection $\mathbb{F}_{q^m}\cong \mathbb{F}_{q}^m$ with the map $x\mapsto (p_1(x),\dots,p_m(x))$
Let $q$ some prime power. Now take the field $\mathbb{F}_{q^m}$.
We need to find polynomials $p_1(x),p_2(x),\dots,p_m(x)\in \mathbb{F}_{q^m}[x]$ such that $\deg{p_i(x)}=q^{m-1}$ and they also satisfy ...
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Number of solutions to $x^2−y^2=a$ over a finite field.
Consider F is a finite field with $ q$ elements, where $q=p^n$, p prime, and n is an integer. Give the number of solutions of pairs (x,y) for the equation $$ x^2-y^2=a $$ according to the value of $ q ...
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Roots of a irreducible polynomial are linearly independent over a finite field.
Question. When does a irreducible polynomial contains linearly independent roots over a finite field?
Motivation. For a finite cyclic Galois extension $E/F$, if $\alpha\in E$ generates a normal basis, ...
- 2,503
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64
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Fields of characteristic $p$ where there exists an element $a$ that is not a p:th power.
Suppose $K$ is a field of characteristic $p$, but that $K \neq K^p$, that is, there exists an element $a \in K$ such that there is no $b \in K$ so that $b^p = a$. We want to prove that this implies ...
- 1,276
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Factorize $x^3-1$ in $F_3[x]$ into irreducible polynomials
I'm fairly new to this topic:
We can see that $x^3-1=(x-1)(x^2+x+1)$
But what I don't understand is how $(x^2+x+1)$ can be further reduced to $(x-1)^2$
Many thanks
- 109
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Partial fraction decomposition over $\mathbb{F}_{n}[x]$
Consider the rational function $\dfrac{f(x)}{g(x)} \in \mathbb{F}_{n}[x]$ such that $g(x) = p(x) \cdot h(x)$. If $\gcd\left(p(x), h(x)\right) = 1$, then $$\dfrac{f(x)}{g(x)} = \dfrac{f(x)}{p(x) \cdot ...
user1236748
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A faster way to check irreducibility of quartics in finite fields
A quadratic or a cubic can be shown to be irreducible in $\mathbb{F}_p$ by showing that none of $0, \dots,p-1$ are roots (or more generally all the elements of $\mathbb{F}_{p^n}$). This does not work ...
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Does $\gcd(x^{p^2+p+1}-1, (x+1)^{p^2+p+1}-1)\neq 1$ in $\mathbb{F}_p[x]$ for all primes $p$?
It seems that for every prime $p$, $x^{p^2+p+1}-1$ and $(x+1)^{p^2+p+1}-1$ are not coprime in $\mathbb{F}_p$. In other words, it seems that there is always a $(p-1)$-th power $x$ in $\mathbb{F}_{p^3}^\...
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Why can't individual terms of a summation not cancel each other in the 2nd case?
Below is from a paper.
$F(.)$ is a low-degree multivariate polynomial over $\mathbb F$ in $s$ variables.
Checking if $\sum_{x \in \lbrace0,1\rbrace^s} F(x) = 0$ will not prove that that $F(x) = 0\...
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