All Questions
Tagged with polynomials finite-fields
171
questions with no upvoted or accepted answers
7
votes
0
answers
167
views
Algebra of additive polynomials
Let $\mathbb{F}_p$ be a finite field for an odd prime $p$. Consider the ring
$$\mathcal{L}(\mathbb{F}_p(X),Y)$$ of additive (or linearized) polynomials in $Y$ over the rational function field $\mathbb{...
- 2,517
6
votes
0
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113
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Factorization of certain polynomials over a finite field
Suppose we have the polynomial $$F_s(x)=cx^{q^s+1}+dx^{q^s}-ax-b \in \mathbb{F}_{q^n}[x]$$ where $ad-bc \neq 0$. The fact that $ad-bc \neq 0$ means that we can take the coefficients of $F_s(x)$ as ...
- 61
6
votes
1
answer
299
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Proof that $t^8+2t^6+4t^4+t^2+1$ is reducible in $\mathbb{F}_p$
Prove that the polynomial $t^8+2t^6+4t^4+t^2+1$ is reducible in $\Bbb F_p$, for all $p\in \Bbb P$.
Here are some examples:
$t^8+2t^6+4t^4+t^2+1=(1 + t + t^4)^2\pmod{2}$
$t^8+2t^6+4t^4+t^2+1=(1 + t) (2 ...
- 15.9k
5
votes
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answers
109
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Does there exists an irreducible polynomial of a given form?
Let $q$ be a prime power, $n$ a positive integer, and $f\in \mathbb F_q[x]$ be an irreducible polynomial. Does there exists $g \in \mathbb F_q[x]$ of degree $n$ such that $f(g)$ is irreducible?
I can ...
- 1,128
5
votes
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469
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Composition of polynomials over finite fields
Consider the set of polynomials of degree at most $n$ over a finite field $k_q$ with $q$ elements where $q$ is prime:
$$
P_{n,q} = \left\{ x + c_2 x^2 + \cdots + c_n x^n:\ c_i \in k_q \right\}.
$$
It ...
- 765
4
votes
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54
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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 $$\dfrac{f(x)}{g(x)} = \dfrac{f(x)}{p(x) \cdot ...
user1236748
4
votes
0
answers
39
views
Zeta function for Powerful Polynomials over Finite Field
I am currently working on a problem that requires me to get find a simpler expresion for:
$$\sum_{f \in \mathcal{S}_h} \frac{1}{|f|^s} $$
Where $\mathcal{S}_h$ is the set of $h$-full polynomials (i.e. ...
4
votes
0
answers
111
views
Compute Galois group of extension
Let $C$ be an algebraically closed field with characteristic $p > 2$
What is the Galois group of the splitting field for the polynomial
$F(X,T) = X(X+1)^p(X-1)^p - T$ over the field of formal ...
- 309
4
votes
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answers
69
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What are the Characteristics of the Ring $\mathbf F_{p}[X]/g$ (mod $g$, $g$ is non-irreducible)?
$f(x)$ is a primitive polynomial of degree $m$ over a field $\mathbf F_{p^m}$. As such, $\mathbf F_{p^m}[x]/f(x)$ forms a ring, with addition and multiplication ($\mkern-4mu\bmod f(x)$), and ...
- 141
4
votes
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answers
110
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Moments of the number of roots of polynomials over finite fields
Let $F:=\{f\in\mathbb{F}_q[X_1,\ldots,X_n]: \textrm{deg}(f)\leq d\}$ be the set containing all $n$-variate polynomials of degree less than or equal to $d$ over finite field $\mathbb{F}_q$ of prime ...
- 51
4
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1k
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How Do I Apply the Cantor-Zassenhaus algorithm to $\mathbb{F}_2$?
Recently, I've been trying to implement the Cantor-Zassenhaus algorithm in C++ over $\mathbb{F}_2$. According to this lecture, the algorithm is basically:
Input is polynomial $f\in\mathbb{F}_q$ with ...
- 436
3
votes
0
answers
67
views
Is this connection between prime numbers, prime polynomials, and finite fields true?
I recently learned of the following connection between prime numbers and prime polynomials in the field of cardinality $2$. Namely, you take a natural number $n$, and use the digits of the base $2$ ...
- 21.4k
3
votes
0
answers
109
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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
3
votes
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674
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Let $p > 2019$ be a prime no. Consider the polynomial $f(x) = (x^2-3)(x^2-673)(x^2-2019)$ How many roots can $f$ possibly have in finite field $F_p$
Let $p > 2019$ be a prime number. Consider the polynomial $f(x) = (x^2-3)(x^2-673)(x^2-2019)$ How many roots can $f$ possibly have in finite field $F_p$ ?
$0$
$2$
$3$
$6$
I have seen this ...
- 513
3
votes
0
answers
113
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Irreducible factorization over $\mathbb{Q}$
I have implemented Berlekamp-Hensel algorithm that provide to factor (into terms of irreducible polynomials) primitive and square-free polynomial over $\mathbb{Z}$. At first, this algorithm obtains ...
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