All Questions
Tagged with polynomials irreducible-polynomials
1,518
questions
6
votes
0
answers
70
views
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 ...
- 5,038
4
votes
2
answers
159
views
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 ...
- 145
0
votes
1
answer
26
views
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 $\...
- 730
2
votes
1
answer
81
views
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
answers
33
views
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
32
views
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 $f(x)$ ?...
- 2,569
3
votes
1
answer
76
views
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
123
views
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
-1
votes
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
68
views
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
0
votes
0
answers
76
views
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^...
- 6,790
1
vote
0
answers
86
views
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
69
views
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 ...
- 510
0
votes
0
answers
37
views
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
3
votes
1
answer
71
views
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 ...
- 521