All Questions
Tagged with polynomials irreducible-polynomials
1,520
questions
-2
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answers
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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
votes
0
answers
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
5
votes
4
answers
206
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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
0
votes
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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votes
1
answer
83
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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
votes
0
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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
119
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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
0
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
1
vote
0
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86
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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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0
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40
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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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votes
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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1
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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
1
vote
0
answers
31
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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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votes
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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votes
0
answers
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
answer
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
191
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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
48
views
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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2
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71
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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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0
answers
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, ...
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4
votes
0
answers
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
answer
53
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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
...
0
votes
0
answers
58
views
Missing and alternating coefficients of polynomials
I will start my question by providing some necessary context:
Let $g(x) = c_lx^l + ... + a_1x + a$ be a polynomial of degree $l$.
$g$ is said to have no missing coefficients if $c_i \neq 0$ for all $...
2
votes
2
answers
63
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If $p(x) = a_3 + a_2x + a_1x^2 + a_0x^3$ is irreducible in $\mathbb{Q}[x]$, then $q(x) = a_0 + a_1x + a_2x^2 + a_3x^3$ is also irreducible?
I have this doubt because what I want to prove is that, with the given hypothesis,$\frac{\mathbb{Q}[x]}{\langle q(x) \rangle}$ is an integral domain. I have the same problem but for polynomials of ...
2
votes
1
answer
156
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Are two principal ideals are equal if their generator has the same root in some extension?
I recently came across the following claim in a paper concerning polyonomials in $\mathcal{R}_p = \mathbb{Z}_p[X]/\langle X^d + 1\rangle$ (where $\mathbb{Z}_p = \mathbb{Z}/\mathbb{Z}p$) and their ...
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0
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Reducing an inseparable polynomial over the same field to a separable polynomial over a field
Description: Let $F$ be a perfect field and $p(x)$ a polynomial over $F$ with multiple roots. Show that there is a polynomial $q(x)$ over $F$ whose distinct roots are the same as the distinct roots of ...
5
votes
1
answer
113
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Irreducible $p(x) \in \mathbb{Q}[X]$ with roots $r, s$ such that $rs = 1$.
If $p(x) \in \mathbb{Q}[X]$ is irreducible and has two roots $r,s$ such that $rs = 1$, then $p(x)$ is of even degree.
I'm not sure how to solve this problem. My initial idea was to consider the field ...
- 464
1
vote
0
answers
36
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divisibility in polynomial ring: if $(X - a)^t | f(X)$ and $(X - b)^t | f(X)$, then does $(X - a)^t \cdot (X - b)^t \mid f(X)$? [duplicate]
let $K$ be a field an $A$ a commutative $K$-algebra with unit. fix an element $f(X) \in A[X]$, unequal elements $a$ and $b$ of $K \subset A$, and an integer $t > 0$.
Question. if $(X - a)^t \mid f(...
- 611
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0
answers
20
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Irreducibility of random polynomials with coefficients distributed according to some probability measure
Theorem 3 in the paper of Bary-Soroker et al. (https://arxiv.org/pdf/2007.14567.pdf) states the following about probability of a polynomial over $\mathbb{Z}$ being irreducible:
Let $H \ge 3$ and $n \...
- 33
1
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0
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73
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Rational polynomial $f$ of minimal degree such that $f(x^{d-1},y^{d-1},z^{d-1})$ is divisible by $x^d+y^d+z^d$
$d\in\Bbb Z^{\ge2}$.
What is the minimal degree of $f\in\mathbb{Q}[x,y,z]$ such that $f(x^{d-1},y^{d-1},z^{d-1})$ is divisible by $x^d+y^d+z^d$?
Let $\zeta=\exp(\frac{2\pi i}{d-1})$.
Since $f(x^{d-1},...
- 3,047
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votes
2
answers
79
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Number of real roots of $f(x) = (x^6) + 2(x^4) + (x^2) - 2(x) + 1$
The question is to find the number of real roots of the polynomial -
$f(x) = (x^6) + 2(x^4) + (x^2) - 2(x) + 1$
I used the Descartes rule of signs. Using it, it is clear that there are maximum number ...
1
vote
1
answer
56
views
A polynomial in two variables is irreducible
Consider the ring of polynomials $\mathbb R[x,y]$ and $I$ the ideal $I=(1+x^2)$ generated by $x^2+1$. I want to determine whether $y^2+1+I$ is irreducible in $\mathbb R[x,y]/(I)$.
If $y^2+1+I$ were ...
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votes
0
answers
61
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A polynomial in $\mathbb Z[x]$ not generated by $5$ and $x^2+2$.
I want to show that the ideal of $\mathbb Z[x]$ generated by the polynomials $5$ and $f(x)=x^2+2$ is strictly included in $\mathbb Z[x]$ so I wanted to find a polynomial $P\in \mathbb Z[x]$ that ...
- 11.1k
0
votes
2
answers
83
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Polynomial factorisation over quotient ring
Let $f(x) = x^4 + x^2 + x + 1 \in \mathbb{Z}_{3}[x] $
Show that $f$ is irreducible over $\mathbb{Z}_{3}$, then factor $f$ over $K = \frac{\mathbb{Z}_{3}[x]}{(f(x))}$.
I already showed that f is ...
- 93
1
vote
1
answer
72
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Polynomial irreducibility over $\mathbb{Z}_{3}[x]$
Let $f(x) = x^4 + x^2 + x + 1 \in \mathbb{Z}_{3}[x]$
Show that $f$ is irreducible over $\mathbb{Z}_{3}$, then factor $f$ over $K = \frac{\mathbb{Z}_{3}[x]}{(f(x))}$
I first tried to use Eisenstein ...
- 93
2
votes
1
answer
155
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$f(X) = X^9 - 2dX^6 + 3d^2 X^3 - d^3$ irreducible in $\mathbb{Q}[X]$ for given cube-free integer $d>1$.
Here's what I've tried:
I've checked that there's exactly one real root for this polynomial (that's easy to prove i.e. there's at least one real root since it has odd degree and it is easy to show ...
1
vote
2
answers
94
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Is $y^{7}+x^{3}z^{4}-x^{2}yz^{4}$ irreducible in $\mathbb{C}[x,y,z]$?
During an exam I had this morning, there was this polynomial $$y^{7}+x^{3}z^{4}-x^{2}yz^{4}$$ and at some point I asked myself if it was irreducible (in $\mathbb{C}[x,y,z]$).
I have to say that the ...
- 169
1
vote
1
answer
68
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Irreducible polynomials in $\mathbb F_q[T]$
Let $q$ be a power of a prime $p$. Is there an infinite set $S$ of $\mathbb N$ such that for every $l\in S$, the polynomial $T^{q^l}-T-1$ is irreducible in $\mathbb F_q[T]$.
It looks like Artin-...
- 1,157
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1
answer
93
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Factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$
How do I factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$?
I checked that the discriminant $D = 16 -4 = 12$ is not a square ($12^{14} = -1 \mod 29$) so this polynomial has no roots. Therefore it's ...
- 2,636
6
votes
6
answers
446
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Is $X^4-3X^2+2X+1$ irreducible over the rationals?
I want to check whether the following polynomials is irreducible over the rationals $f=X^4-3X^2+2X+1$. I think I found that it is irreducible but my solution is really complicated, quite long and I am ...
- 101
0
votes
0
answers
59
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Question about reduction mod p method for polynomials [duplicate]
I am a undergradute math student and I was traying to prove the polynomial $p(x)= x^4-10x^2+1$ is irreducible in $\mathbb{Q}[x]$. I used reduction mod p and for every prime p I found a factorizarion ...
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0
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1
answer
101
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Prove that $6x^5-55x^3 + 50x^2+15$ is irreducible over $\mathbb{Q}[i]$ [duplicate]
We need to prove that $6x^5-55x^3 + 50x^2+15$ is irreducible over $\mathbb{Q}[i]$. We let $f(x) = 6x^5-55x^3 + 50x^2+15$. We use Eisenstein's criterion. We know that $(5)$ is a prime ideal in $\mathbb{...
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