All Questions
21
questions
0
votes
3
answers
120
views
Requesting suggestion for software to solve polynomial equation with coefficients in GF(1024)
For example, the equation $x^2 + x + 1 = 0$ has the solutions $\alpha^{682}$ and $\alpha^{341}$, where $\alpha$ is a primitive element of GF(1024). I am looking for any software that can solve these ...
- 19
0
votes
1
answer
405
views
Irreducible polynomial in integers modulo p
I am a completing a past paper question and I am undecided on what method to use here. The question is:
For what $a$ is $f(x)=x^3+x+a\in\mathbb{Z}_{7}[x]$ irreducible? My ideas are:
(1) Check each $a\...
- 99
1
vote
0
answers
35
views
Software to solve system equations over finite field extension.
I am working with equations in a finite field extension and I am looking for a mathematical software to solve the following problem.
Let $k$ be a field $GF(4)$. I have a function $f:k^2 \to k^2$ ...
- 165
2
votes
1
answer
50
views
Finding ordered pair of polynomial $(f(x),g(x))$ with coefficient $\mathbb{Z}_{p}$ for a prime number $p$
In process of solving a problem, I had to find some ordered pair of polynomial $(f(x),g(x))$ with coefficient $\mathbb{Z}_{p}$, $p$ is prime number. (i.e $f(x),g(x) \in \mathbb{Z}_{p}[x])$
such that
$\...
- 29
2
votes
0
answers
60
views
Generator matrix of twisted Gabidulin codes
If we consider twisted Gabidulin codes proposed by Sheekey as follows:
Let $n, k, s$ be positive integers such that $k<n$ and $\gcd(s, n)=1$. Let $\eta$ be a nonzero element in $\mathbb{F}_{q^...
- 51
2
votes
1
answer
90
views
A problem about irreducible polynomials over a finite field.
Problem
Let $F$ be a finite field with $q$ elements. Let $f\in F[x]$ be an irreducible polynomial. Prove that if $f \mid x^{q^n}-x$ then $\deg{f}\mid n$ (the converse is also true and I have a proof).
...
- 1,336
1
vote
0
answers
279
views
Splitting field of $x^5 - t$ over $\mathbb{F}_{11}(t)$
I'm trying to solve a question which asks me to find the degree of the splitting field of $x^5 - t$ over $\mathbb{F}_{11}(t)$.
I honestly have no idea where to begin - the only other examples of ...
- 11
2
votes
2
answers
170
views
Factorization of a polynomial in $\Bbb F_7$
I need to reduce as much as possible the polynomial: $x^6+3x^5+2x^4+6x^3+4x^2+5x+2$ in the finite field $\Bbb F_7$.
It has no roots over the field, and I don't think that it is necessary to check ...
- 195
3
votes
1
answer
175
views
How does a field extension maintain the field structure, and are all field extensions fields, or only algebraic extensions?
Suppose we adjoin a symbol $k$ to the field $F_p$, as in $F_p(k)$. What is an intuitive understanding of the structure of this field and its elements? Since multiplication needs to be closed, all new ...
- 2,057
1
vote
2
answers
225
views
Is $\phi: \mathbb{F_p}(X) \rightarrow \mathbb{F_p}(X), a \mapsto a$ for $a \in \mathbb{F_p}$ and $X \mapsto X+1$ a field homomorphism?
Let $\mathbb{F_p}$ be a finite field of cardinality $p$, $p$ a prime. Let $\phi: \mathbb{F_p}(X) \rightarrow \mathbb{F_p}(X), a \mapsto a$ for $a \in \mathbb{F_p}$ and $X \mapsto X+1$, aka $\phi$ ...
- 1,517
0
votes
2
answers
760
views
Construct a extension field over $Z_2$
I have an polynomial $x^4+x+1 \in \mathbb{Z}\left\{ x\right\}$ and I want to construct an extension field of $\mathbb{Z}_2$ that include the roots of that polynomial. So is this the right approach?
...
- 135
0
votes
1
answer
83
views
How to find the minimal polynomial of a polynomial over a finite field
Let $\mathbb{F}_{p^n}$ be a finite field and $f(x) \in \mathbb{F}_{p^n}[x]$ which is monic and non-constant. My question is:
$i)$ Is there the minimal polynomial $g(x) \in \mathbb{F}_{p}[x] \subset \...
- 773
1
vote
1
answer
614
views
How to reduce a polynomial over an extension field (with or without MAGMA)
I have the elements $u^{4770}$ and $u^{7489}$ lying in the finite field $\mathbb{F}_{29^3} = \mathbb{F}_{29}[u]$ where $u^3+2u+27=0$. I'd like to find equivalent values with lower degrees so that I ...
- 1,862
2
votes
1
answer
138
views
If $a$ root of $h(x)\in\mathbb{F}_q[x]$ and $a$ in the extension field $\mathbb{F}_Q$ of $\mathbb{F}_q$ then $a^{q^r}$, $r\geq0$ are also roots
Let $\mathbb{F}_Q$ be an extension field of $\mathbb{F}_q$. Show that if an element $a\in \mathbb{F}_Q$ is a root of a polynomial $h(x)\in\mathbb{F}_q[x]$, then so are the elements $a^{q^r}$ for $r\...
- 1,381
3
votes
0
answers
63
views
Determining which finite fields a multi-variant polynomial has roots
Suppose $K$ is a finite field and $q \in K[x_1,\dots,x_n]$. In general, is there anything we can say about which extension fields of $K$ the polynomial $q$ has roots?
In the special case where $n = ...
- 186