All Questions
111
questions
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
1
vote
1
answer
63
views
Factorise $x^5-x$ in $\mathbb{C}[x]$ and $\mathbb{R}[x]$
I would like general feedback on my solution to this exercise.
Exercise
Factorise $x^5-x$ in $\mathbb{C}[x]$ and $\mathbb{R}[x]$.
Solution
In $\mathbb{C}[x]$ we have
\begin{align*}
&\phantom{=}\,\,...
- 41
0
votes
1
answer
212
views
Show that $3x^5-4x^3-6x^2+6$ is irreducible over $\mathbb{Q}[x]$
Can’t seem to apply Eisenstein’s criterion. It doesn’t make sense to reduce modulo 2 since the constant term disappears… What are some other possible strategies to apply here?
- 91
2
votes
1
answer
64
views
Is there a 3rd degree irreducible polynomial over Q[x], such that two of it's roots' (over C[x]) product equals the third root?
So we have a polynomial in the form: $ax^3+bx^2+cx+d$, where $a,b,c,d\in\mathbb{Q}$, $a\neq 0$. And this is irreducible over $\mathbb{Q}[x]$, but is of course reducible over $\mathbb{C}[x]$.
We have ...
- 29
15
votes
3
answers
642
views
Under the which condition, factorisation of $a_1^n+a_2^n+\cdots+a_n^n-na_1a_2a_3...a_n ?$ is possible?
Under the which condition, factorisation of the polynomial
$$a_1^n+a_2^n+\cdots+a_n^n-na_1a_2a_3...a_n ?$$
is possible?
I know possible cases:
$$a^2+b^2-2ab=(a-b)^2$$
and
$$a^3+b^3+c^3-3abc=(a+b+c)(a^...
- 1,659
-2
votes
2
answers
99
views
Show that the polynomial $P(x)=x^4-x^2-x+2$ has no real roots [closed]
Using clever algebra show that the polynomial
$$P(x)=x^4-x^2-x+2$$
has no real roots.
Obviously, we can not use the derivative.
Using the general quartic formula is terrible.
I tried
$$(x^2+1)^2-3x^...
- 652
0
votes
1
answer
80
views
Reduce the degree and solve the polynomial equation $x^6+ax^4-2x^3+1=0$ by algebraic tricks
Reduce the degree of the polynomial and solve by algebraic tricks: $$x^6+ax^4-2x^3+1=0$$ where $a\in\mathbb R$.
$a=0$ is obviously trivial. I tried all possible algebraic variations.
$$\frac {P(x)}{x^...
- 652
3
votes
3
answers
174
views
Find the all real roots of the polynomial $x^6+3 x^5+3 x-1=0$ in closed form
Find the all real roots of the polynomial
$$x^6+3 x^5+3 x-1=0$$
in exact form.
WolframAlpha gives only numerical results. I've asked a few similar questions before. The source of the problem comes ...
- 652
2
votes
1
answer
116
views
Solve the polynomial $x^6+x^5+x^4-2x^3-x^2+1=0$ in exact form
Try to reduce the degree of the polynomial $$P(x)=x^6+x^5+x^4-2x^3-x^2+1$$ by algebraic ways and find the possible solution method to $P(x)=0$.
The source of the problem comes from a non-english ...
- 652
2
votes
2
answers
314
views
Maximal degree of irreducible polynomials
This is a question I have thought about for a while. We know that every polynomial $p \in \mathbb C[z]$ can be written as a product of monomials
$$p(z) = a \displaystyle\prod_{i=1}^n(z-z_i).$$
Now for ...
- 1,397
2
votes
1
answer
54
views
Equation with polynomial with integer coeficients.
Let $p>3$ be a prime number.
Prove that there doesn't exist a pair of polynomials $(f,g)\in{\mathbb{Z}[X]\times\mathbb{Z}[X]}$ such that:
$X^{2p}+pX^{p+1}-1=[(X+1)^p+p\cdot f(X)]\cdot[(X-1)^p+p\...
- 1,144
1
vote
0
answers
256
views
Factorization and roots of a multivariate polynomial over finite fields
I am interested to know whether factorization of a multivariate polynomial $f(x_1, x_2, \dots, x_n) \in \mathbb{F}_p[x_1, x_2, \dots, x_n]$ into irreducible factors yields some information about the ...
- 165
1
vote
1
answer
62
views
Polynomial factorisation problem
I was wondering if the following statement is correct:
Let $P(x,y)$ be a real polynomial, and $\gcd(r,s)=1$. If $y-x^s$ divides $P(x^r,y)$, then $y^r-x^{rs}$ divides $P(x^r,y)$ (both in the real field)...
- 2,015
1
vote
1
answer
91
views
Every polynomial with constant term 1 can be factorized using degree one polynomials of the for (1-ax)
I've seen a statement along the lines of:
If $K$ is an algebraically closed field, then every polynomial $P$ with $P(0) = 1$ in $K[x]$ can be expressed as $\Pi_{1 \leq i \leq n} (1 - \lambda_i x)$.
...
- 1,044
3
votes
1
answer
164
views
When does $x^n-n$ factor?
Suppose that $n=(kj)^k$ for some integers $j\geq 1$ and $k\geq 2$. Then writing $m=k^{k-1}j^k$, we have
$$x^n-n=(x^m)^k - (kj)^k$$ and so $x^m-kj$ is a factor of $x^n-n$. I would like to show that ...
- 1,136