All Questions
39
questions
4
votes
1
answer
99
views
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}^\...
- 1,923
0
votes
1
answer
143
views
Factorization of $x^{38} - 1$ in $F_5$
I am looking for a way to factorize $x^{38} - 1$ polynomial in $F_5$ . So far I have:
$x^{38} - 1 = (x^{19} - 1)(x^{19} + 1) = (x - 1)(x^{18} + x^{17}... + 1)(x + 1)(x^{18} - x^{17} + ... + 1)$
I can ...
8
votes
2
answers
267
views
How to factor a polynomial quickly in $\mathbb{F}_5[x]$
I was doing an exercise in Brzezinski's Galois Theory Through Exercises and needed to factor the polynomial
$x^6+5x^2+x+1=x^6+x+1$ in $\mathbb{F}_5[x]$. Is there a quick way to do this?
I can see it ...
- 11k
0
votes
1
answer
69
views
Factor the polynomial $x^{15}+3x^{10}+2x^5+4$ in $\mathbb{F}_5[x]$.
Factor the polynomial $f(x)=x^{15}+3x^{10}+2x^5+4$ in $\mathbb{F}_5[x]$.
Alright, so I'm sure this is a simple answer but I have not been able to figure out, hopefully I can quickly be set on the ...
- 2,454
1
vote
0
answers
256
views
Factorization and roots of a multivariate polynomial over finite fields
I am interested to know whether factorization of a multivariate polynomial $f(x_1, x_2, \dots, x_n) \in \mathbb{F}_p[x_1, x_2, \dots, x_n]$ into irreducible factors yields some information about the ...
- 165
0
votes
1
answer
50
views
factoring a polynomial in $\mathbb{Z}_3[x]$
I asked this question. Now I am trying to factor the same polynomial, $f(x)=x^5+2x^4+3x^3+5$, in $\mathbb{Z}_3[x]$.
So, we have that $f(x)=x^5+2x^4+2$ in $\mathbb{Z}_3[x]$. Like in my previous ...
- 700
1
vote
0
answers
90
views
Factorizing polynomials over finite fields and $\mathbb{Q}$
I have the following problem:
Factor the integer polynomial $x^5+2x^4+3x^3+5$ modulo 2, and over $\mathbb{Q}$.
For $2$, the polynomial becomes $f(x)=x^5+x^3+1$. It is easy to check if it has roots ...
- 700
1
vote
2
answers
239
views
Proof of square-free polynomial factorization over finite fields
Why should it be that the greatest common divisor of a polynomial and it's derivative produce the product of the repeated factors?
What about the case where the derivative is zero in $\mathbb{F}_p$? ...
- 159
0
votes
1
answer
725
views
Factoring $x^{n}-1$ over the finite field $\mathbf F_q$.
If $\mathbf F_q$ is a finite field. Then how to factor $x^{n}-1$ in $\mathbf F_q[x]$?
I think we can suppose that $(q,n)＝1$,since if $n＝q^{e}n'$ then $x^{n}-1＝(x^{n'}-1)^{q^{e}}$ so that we just ...
- 11
3
votes
1
answer
79
views
Solution Verification: Factoring $\left|\begin{smallmatrix}x&y&z\\x^p&y^p&z^p\\x^{p^2}&y^{p^2}&z^{p^2}\end{smallmatrix}\right|$ over $\mathbb{Z}_p.$
Problem: Factor $\begin{vmatrix} x & y & z \\ x^p & y^p & z^p \\
x^{p^2} & y^{p^2} & z^{p^2} \end{vmatrix}$ over $\mathbb{Z}_p$ as a
product of polynomials of the form $ax+by+...
- 5,230
2
votes
1
answer
52
views
Factoring a polynomial over $\mathbb{F}_{101}$.
I am trying to show that the polynomial $f = X^5 + 69$ factors in linear factors over $\mathbf{F}_{101}$. I have found one root, namely $2$, since $2^5 = 32$, and so $2^5 + 69 = 101 = 0$. But I cannot ...
- 1,659
4
votes
3
answers
1k
views
Show that $x^6+x^3+1$ is irreducible over $\mathbb{F}_2$
This is part of a larger question where I'm supposed to determine if $x^6+x^3+1$ is irreducible over $\mathbb{F}_2$, $\mathbb{F}_3$, $\mathbb{F}_{19}$ and $\mathbb{Q}$.
For $\mathbb{F}_3$ and $\...
- 43
0
votes
2
answers
174
views
How can the polynomial $x^7+1$ be factored in $\mathbb F_2$? [duplicate]
I want to factorize this polynomial $x^{7}+1$.
The result that I expect is $(x+1)(x^{3}+x+1)(x^{3}+x^{2}+x+1)$
What is the best way to proceed?
As it seems the factorization is conducted in $\...
- 183
4
votes
1
answer
983
views
Factorising $X^{16}- X$ over $\mathbb F_4$.
I need to factorise $X^{16}- X$ over $\mathbb F_4$. How might I go about this? I have factorised over $\mathbb F_2$ and I know the quadratic must split but I'm not sure about the quartic and octic. Is ...
- 297
2
votes
2
answers
41
views
Factoring $2t^4+t^2+2$ into prime polynomials in $\Bbb F_3$
I want to factor $2t^4+t^2+2$ into prime polynomials in $\Bbb F_3$.
I know that in $\Bbb F_3$, we have $2=-1$ so $2t^4+t^2+2=-t^4+t^2-1$ but I can't get any further.
Thanks for your help!
- 23