All Questions
Tagged with polynomials irreducible-polynomials
1,520
questions
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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 $\mathbb{Z}[x]$ ?
6
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0
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82
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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
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votes
4
answers
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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 ...
- 155
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1
answer
27
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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 $\...
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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
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35
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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
0
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 $f(x)$ ?...
- 2,589
3
votes
1
answer
78
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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, ...
- 3,940
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 ...
- 637
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2
answers
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How to prove: $f\equiv0\,(\mathrm{mod}\,p^2)\iff f'\equiv0\,(\mathrm{mod}\,p)$? [closed]
Edit: Corrected the mod order.
It might be trivial, but I have no idea at all about it.
For a univariate polynomial $p$, then $f\equiv0\,(\mathrm{mod}\,p^2)\iff f'\equiv0\,(\mathrm{mod}\,p)$
where $f'$...
- 1,329
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votes
0
answers
69
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Reducibility of $x^2-7$ over $\mathbb{Q}(\sqrt[5]{3})$
Suppose for a contradiction that $x^2-7$ is reducible over $\mathbb{Q}(\sqrt[5]{3})$. Then $\sqrt{7}\in\mathbb{Q}(\sqrt[5]{3})$. It follows that $\mathbb{Q}\subset\mathbb{Q}(\sqrt{7})\subset\mathbb{Q}(...
- 2,066
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0
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76
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Showing that $x^4+2x^2+5$ is irreducible over rational numbers [duplicate]
I want to show that $P(x)= x^4+2x^2+5$ is irreducible over rational numbers. I have decomposed the polynomial into $(x^2+ax+b)(x^2+cx+d)$, and since $P(x)$ is an even function, we have either $P(x)=(x^...
- 6,794
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Number of irreducible polynomials of degree at most n over a finite field
We know that the number $N(n,q)$ of irreducible polynomials of degree $n$
over the finite field $\mathbb{F}_q$
is given by Gauss’s formula
$$N(n,q)=\frac{q-1}{n} \sum_{d\mid n}\mu(n/d)q^d.$$
The number ...
2
votes
1
answer
70
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If $f(x)\in \mathbb{Z}[x]$ is irreducible (over $\mathbb{Q}$), is it always possible to find $a$ and $b$ in $\mathbb{Q}$ with $f(ax+b)$ Eisenstein? [duplicate]
My initial thought is no, simply because it seems too easy if it is true.
The simplest example of a nontrivial irreducible polynomial I could think of was $f(x)=x^2+1$. Unfortunately, $f(x+1)$ is ...
- 520
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Artin's theorem exercise - proving that the fixed field is generated by the coefficients of the minimal polynomial
Suppose $L/K$ is a finite extension. $G$ is a finite group of $K$-automorphisms of $L$. Denote by $L^G$ the field elements of $L$ fixed by action of $G$. For any $\alpha \in L$ we write $f(t, \alpha) =...
- 474
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1
answer
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For what integers $m\gt n\gt 0$, the polynomial $x^m+x^n+1$ is irreducible over $\mathbb Q$?
I came up with this problem and have found it interesting.
Problem. For what integers $m\gt n\gt 0$, the polynomial $f(x)=x^m+x^n+1$ is irreducible in $\mathbb Q[x]$?
If $mn\equiv 2 \pmod 3$, i.e. one ...
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2
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Is this $f(x) = x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$ irreducible in GF(5)?
Perhaps one can somehow apply Eisenstein's sign here by considering $f(x+1)$, but by default it is formulated for the expansion over $\mathbb{Q}$ of a polynomial from $\mathbb{Z}[x]$. Here we have $GF(...
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Prove: either there exists a $ j $ such that $ |x_j| > 1 $, or $ |x_j| = 1 $ for all $ j $.
Let $ f(x) = (x - x_1) \cdots (x - x_n) $ be an $ n $-degree monic irreducible polynomial with integer coefficients. Prove: either there exists a $ j $ such that $ |x_j| > 1 $, or $ |x_j| = 1 $ for ...
- 11
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Find a monic irreducible polynomial equivelent to $(x-x_1)(x-x_2)\Phi_m$
Find a monic irreducible polynomial $f(x) = (x - x_1) ... (x - x_n)$， $|x_1| > 1$ and $x_1$ is real, |x_2| < 1 and
$x_2$ is real, $|x_j| = 1$ for all $j > 2$. And First, prove $n > 3$ ...
- 11
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0
answers
54
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When is $f = X^4 -1 \in \mathbb{F}_p[X], p $prime, irreducible and/or seperable? [duplicate]
I'm having some trouble figuring out a solution to this. I understand that $f$ is separable, iff all its roots are distinct, however I'm completely clueless about how to investigate that criterion......
- 17
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64
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Solving sextic with Kampé de Fériet functions
I recently faced a problem with a polynomial of 6th degree, a sextic. I want an analytical solution to the problem, and I read in the last few days that Kampé de Fériet functions can solve general ...
0
votes
1
answer
32
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$f$ is irreducible if the polynomial reduced $p$ is irreducible and the degrees are the same
Let $f$ be an irreducible polynomial and $h(f)$ the polynomial with coefficients reduced modulo a prime $p$. Then if $\deg(f)=\deg(h(f))$ and $h(f)$ is irreducible then $f$ is irreducible as an ...
0
votes
1
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45
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Irreducible polynomials with complex root.
I need to show that if $f$ and $g$ are irreducible in $\mathbb{Q}$[$x$] and they share a common complex root, then there is $a \in \mathbb{Q}$ such that $f = a . g$.
What I thought:
Call $u \in \...
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7
votes
1
answer
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Factorization and irreducibilty for $x^n-2x^m+1$ trinomials.
I have encountered a weird phenomenon while trying to solve a problem on Reddit. Here is the phenomenon.
Let $a>b \in \mathbb{N}$ and $p_{(a,b)} = x^a - 2x^b + 1$
It seems that if $gcd(a,b,c,d) = 1,...
- 309
1
vote
1
answer
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is $x^4-x+1$ irreducible in $\mathbb{Z}_3$
i was wondering if i checked correctly. i found all polynomials in $\mathbb{Z}_3[x]$ of degree 2 which are irreducible and checked if they are divisible without remainder
the polynomials i tried were
$...
- 37
6
votes
2
answers
151
views
Irreducible polynomial in $\Bbb{Z}_2[x]$
Suppose $2k + 1 \equiv 3 \mod 4$ in $\Bbb{Z}_{\geq 1}$.
Is the polynomial: $p_k(x) = x^{2k + 1} + x^{2k - 1} \dots + x + 1$ irreducible in $\Bbb{Z}_2[x]$?
I do not know whether it is true or not...
(...
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Is a polynomial monotone when the first derivative has only imaginary roots? [closed]
I have a polynomial over a specific the range. The first derivative has only two imaginary roots and no real roots. The first derivative is positive in the lower bound and upper bound. Does that mean ...
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35
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Showing the polynomial has integer coefficients
Show that $\Phi_n(X)$ has integer coefficients.
The proofs here states that $$\Phi_n(X)=\frac{X^n-1}{\prod_{d|n,d\ne n}\Phi_d(X)}.$$
And by long division, they get $\Phi_n(X)\in \Bbb{Q}[X]$. However, ...
- 1,711
4
votes
0
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74
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Prime ideals in $\mathbb{Z}[x]$ containing $\langle 3\rangle+\langle f\rangle$
This question is from a 2002 Harvard qualifier:
Let $R=\mathbb{Z}[x]/(f)$ where $f(x)=x^4 - x^3 + x^2 - 2x + 4$. Let $I = 3R$ be the
principal ideal of $R$ generated by $3$. Find all prime ideals $\...
- 2,234
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1
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Prove that $f(x)$ is irreducible in $\mathbb{Z}$ with $f(b)$ a prime, $f(b-1) \neq 0$ and $\Re(\alpha_i) < b -1/2$
I need some help with a lemma I need to prove.
First I will provide some background with previous lemmas that I already have been able to prove. Maybe these lemmas are needed to proof the last lemma
...
