All Questions
Tagged with polynomials irreducible-polynomials
1,520
questions
-2
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What is the criteria to find out if a polynomial is irreducible? [closed]
Is the polynomial $3x^3-5x^2+7$ irreducible over $\mathbb{Z}[x]$ ?
3
votes
2
answers
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Factor $x^5-1$ into irreducibles in $\mathbb{F}_p[x]$
I have to factor the polynomial $f(x)=x^5-1$ in $\mathbb{F}_p[x]$, where $p \neq 5$ is a generic prime number.
I showhed that, if $5 \mid p-1$, then $f(x)$ splits into linear irreducible.
Now I ...
- 61.5k
1
vote
1
answer
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Why is this polynomial irreducible?
I have this polynomial:
$$ x^4 - 11x^3 + 27x^2 - 11x -13$$
And I need to check that is irreducible in $\mathbb{Q}[x]$. What I saw is that if it has a factorisation 3-1, it happens that must have a ...
- 95
12
votes
2
answers
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Is the polynomial $x^{105} - 9$ reducible over $\mathbb{Z}$?
Is the polynomial $x^{105} - 9$ reducible over $\mathbb{Z}$?
This exercise I received on a test, and I didn't resolve it. I would be curious in any demonstration with explanations. Thanks!
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votes
4
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Prove that $x^6+5x^2+8$ is reducible over Z (integer)?
$attempts:-$
1] I tried to replace $X^2=t$ but nothing click after that .
2] then I tried to replace this polynomial say P(x) by P(x+1) or P(x-1) to apply Eisenstein's Irreducibility Criterion Theorem ...
- 11.9k
6
votes
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answers
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Is the area enclosed by p(x,y) always irrational?
Take a polynomial $p \in \mathbb{Q}[X,Y]$. Now draw the graph of $p(x,y)=0$. If, like $X^2-Y^2-1$, this turns out to enclose a finite area, is the area enclosed always irrational?
There are some ...
- 5,593
3
votes
3
answers
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Is $x^4+x+1$ irreducible in $\Bbb{Q}[x]$?
Decide with a proof if $f(x)=x^4+x+1$ is irreducible in $\Bbb{Q}[x]$.
I was thinking of using DeMoivre's Theorem but I'm not sure how.
Thanks!
- 73
0
votes
1
answer
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Determining the Equality of Two Field Extensions
Let $F$ be a field of characteristic $0$. Let $F(\alpha)/F$ be a finite extension of degree not divisible by $3$. Is is true that $F(\alpha^3)=F(\alpha)$? If we assume that they are not equal, since $\...
- 82.1k
2
votes
1
answer
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If $F/K$ is normal extension and $f \in K[x]$ irreducible and $f=\prod_{i=1}^{n}g_{i}^{m_{i}}$ in $F[x]$ then $m_{i}=m_{j}$ for all $i,j$
So I have that question:
Let $F/K$ be a normal extension and $f$ irreducible polynomial in $K[x]$ assume that $f=\prod_{i=1}^{n}g_{i}^{m_{i}}$ where $g_i$ is irreducible in $F[x]$ ($m_i \geq 1$) then ...
- 115
0
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Polynomial reduction modulo n. Irreducible polynomal.
I have the following polynomial: $f(x)=x^4+1$. I have to prove that it is irreducible over $\mathbb{Z}[x]$ using reduction criterion.
The Reduction Criterion says that:
Let $\mathfrak{m}$ be maximal ...
- 845
2
votes
1
answer
127
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Irreducibility of a Polynomial with Prime Exponents
Let $f(x) = (x^p - a_1)(x^p - a_2) \ldots (x^p - a_{2n}) - 1$
where $a_i \geq 1$ are distinct positive integers where at least two of them are even, and $n \geq 1$ is a positive integer and $p$ is ...
- 17k
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votes
1
answer
37
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To determine the number of roots for all antiderivative of a cubic polynomial
Let $f(x)$ be a cubic polynomial with real coefficients. Suppose that $f(x)$ has exactly one real root which is simple. Which of the following statements holds for all antiderivative $F(x)$ of $f(x)$ ?...
- 188
3
votes
1
answer
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An efficient algorithm for determining whether a quartic with integer coefficients is irreducible over $\mathbb{Z}$
I'm interested in what efficient algorithm could be used for determining if a quartic polynomial with integer coefficients is irreducible over $\mathbb{Z}$.
For quadratics and cubics it's not too bad, ...
- 41.4k
104
votes
8
answers
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How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?
Let $p$ be a prime. How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?
Right now I'm able to prove that it has no roots and that it is separable, but I have not ...
- 61.5k
3
votes
2
answers
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Analytical solution of a polynomial with non integer order
Can anyone think of a possible analytical solution of the following equation?
$x\left(1-0.2x^2\right)^{5/2}=constant$
I am not a mathematician, but, it seems to me that only numerical methods can ...
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