All Questions
10
questions
1
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0
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42
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If $f(x)=ax^{2p}+bx^p+c\in\mathbb{F}_p[x]$, prove that $f'(x)=0$
If $R$ is a commutative ring, then the set of all polynomials with coefficients in $R$ is denoted by $R[x]$.
$\mathbb{I}_p[m]$ is the integers mod $m$.
When $p$ is a prime, we will usually denote the ...
- 366
-1
votes
1
answer
1k
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What is the inverse of an element of polynomial ring over finite field?
Let's consider the polynomial ring $\mathbb{F}_q[x]$. How to find the inverse of an element of this ring. For example, If I'm working over $\mathbb{Z}_7[x]$, what is the inverse of $x^2+x+1$. This is ...
- 175
1
vote
1
answer
55
views
Let p be prime and Fpm denote the field with pm elements. Let k,n be in N.
I am just trying to solve the first part
I think I understand how to attempt this problem using a specific p, k, and n, my steps are as follows:
We know that the field $\mathbb F_{p^m}$ = the roots ...
- 13
2
votes
0
answers
245
views
Applications of the Hermite's criterion?
I found this statement on permutation polynomials and I was wondering in which domain we can find applications and what is its aim.
Here is the criterion : «If $q=p^n$ with $p$ a prime number then $f\...
- 3,330
1
vote
2
answers
327
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Irreducible polynomials in GF(p) [duplicate]
Given is the polynomial $\varphi(X)=X^4+1$.
Now there are two tasks:
(1) Show, that $ \varphi(X)$ is reducible in $\mathbb F_p [X]$, where $p$ is prime number with $p \equiv1$ (mod 4).
(2) Show, ...
- 834
4
votes
1
answer
241
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Cubic root of a polynomial to modulo of another polynomial
Is there any algorithm to solve problems like the problem below, any ideas?
Find polynomial $f$ such that:
$f^3 \equiv x^4 + x^2 \ (mod\ x^{10} + x^3 + 1)$
All numeric coefficients are from $\...
- 125
0
votes
1
answer
239
views
monic irreducible polynomials
I need help with :
Let $\Bbb F_3=\Bbb Z_3$ be the field with 3 elements.Show that there are infinitely many monic irreducible polynomials in $\Bbb F_3[x]$ such that $P(0)=-1$.
Now,I saw this proof ...
- 2,002
6
votes
2
answers
100
views
Roots of $x^p + x + [\alpha]_p \in \mathbb{F}_p[x]$
Let $$g(x) = x^p + x + [\alpha]_p \in \mathbb{F}_p[x],$$ where $p$ is prime.
For which $\alpha, p \in \mathbb{Z}$ does $g(x)$ have at least
one root? And for which $\alpha, p \in \mathbb{Z}$ does $g(...
user310769
3
votes
2
answers
2k
views
$x^p-x \equiv x(x-1)(x-2)\cdots (x-(p-1))\,\pmod{\!p}$
I got a question to show that :
If $p$ is prime number, then
$$x^p - x \equiv x(x-1)(x-2)(x-3)\cdots (x -(p-1))\,\,\text{(mod }\,p\text{)}$$
Now I got 2 steps to show that the two polynomials ...
- 33
3
votes
1
answer
680
views
Factoring polynomials of the form $1+x+\cdots +x^{p-1}$ in finite field
Suppose $p$ and $q$ primes and $p$ is odd. Then, are there nice and clever ways to factorize polynomials of the form $$f(x)=1+x+\cdots +x^{p-1}$$ in the ring $\mathbb{F}_q[x]$ ? What about the case ...
- 10.2k