All Questions
Tagged with polynomials irreducible-polynomials
284
questions with no upvoted or accepted answers
16
votes
1
answer
710
views
Is $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ an irreducible polynomial over $\mathbb{Q}[X]$?
Let $a, b \in \mathbb{Q}$, with $a\neq b$ and $ab\neq 0$, and $n$ a positive integer.
Is the polynomial $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ irreducible over $\mathbb{Q}[X]$?
I know that $\bigl(X(X-...
- 2,037
10
votes
0
answers
176
views
Irreducibility of q-factorial plus 1
Is it true that $[n]_q! + 1$ is an irreducible polynomial over $\mathbb{Z}$ for all positive integers $n$ ?
I checked that this is true for $n$ up to $20$.
Here $[n]_q! := 1 (1 + q) (1 + q + q^2) \...
- 201
8
votes
0
answers
148
views
If prime $p=a_n10^n+a_{n-1}10^{n-1}+\ldots+a_110+a_0$ then $f(x)=a_nx^n+\ldots+a_0$ is irreducible in $\mathbb{Z}[x]$
I have been trying to solve this problem on my own for four days now, and I cannot figure out how to prove it:
If we express a prime $p$ in base $10$ as
$$p= a_m10^m+a_{m-1}10^{m-1}+\ldots +a_110+a_0,...
- 267
7
votes
0
answers
507
views
Eisenstein's criterion for two variables
I want to know if there is a criterion, like Eisenstein's criterion, for polynomials with 2 variables?
If there is, how does it work?
- 71
6
votes
0
answers
82
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 ...
- 5,593
5
votes
0
answers
99
views
Determine the galois group of this 6 degree polynomial over $\mathbb{Q}$
Determine the Galois group of $f(x)$ over $\mathbb{Q}$
$f(x)=x^6+22 x^5-9 x^4+12 x^3-37 x^2-29 x-15$
This question comes from Johns Hopkins University Fall 2018 algebra qualifying.
I have found it is ...
- 399
5
votes
0
answers
292
views
Family of irreducible polynomials over $\mathbb{Z}$
Note. In this question I am using the following definition of irreducible polynomial: "Polynomial is irreducible if it cannot be factored into polynomials with coefficients in the same domain ...
- 12.9k
5
votes
0
answers
37
views
For which rings $R$ can the number of irreducible factors of degree at least one of $f\in R[X]$ be bounded in terms of $\deg f$?
Question: For which integral domains $R$ does there exist a function $M:\mathbb{Z}_{\ge 0}\to\mathbb{Z}_{\ge 0}$, such that for all $f\in R[X]$, we have that $f$ has at most $M(\deg f)$ irreducible ...
- 8,421
5
votes
0
answers
64
views
Irreducibility of $p(x)$ implies that of $p(x+c)$ only when taken over a field?
$R$ is a ring and $R[x]$ is the polynomial ring over $R$ . $c$ is any fixed element of $R$ . Then the map $f(x)\mapsto f(x+c)$ is an isomorphism from $R[x]$ to itself. Now ...
- 5,897
5
votes
0
answers
348
views
Irreducibility of polynomials via Frobenius map
I am having trouble trying to show this:
Let $f \in \mathbb{F}_p[x]$ be a non-constant polynomial and let $F$ denote the Frobenius map $F: R \rightarrow R$ where $R = \mathbb F_p[x]/(f)$. Prove ...
- 573
4
votes
0
answers
74
views
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
4
votes
0
answers
145
views
A rational function with hidden symmetry and alternating poles and zeros.
Upon answering a question about an equivalence of two binomial sums I have noted that a naturally appearing function has some interesting properties.
Consider the function:
$$
f(m,n_1,n_2;z)=\frac{1}{...
- 26.7k
4
votes
0
answers
1k
views
Proof check: Irreducibility of $xy-1$
Mainly I ask, to check if there are holes in my logic anywhere locally, since these will affect me globally, and perhaps other readers.
If I want to prove that $xy-1\in \Bbb{C}[x,y]$ is irreducible, ...
4
votes
0
answers
1k
views
proving that $8x^3-6x-1$ is irreducible over $\mathbb{Q}$
When considering the impossibility of trisecting the 60 degree angle, one comes across the polynomial $f(x)=8x^3-6x-1$, which I want to prove is irreducible over $\mathbb{Q}$. I reduced the ...
- 1,232
4
votes
0
answers
110
views
Irreducibility of some polynomial
Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
- 1,263