All Questions
15
questions
3
votes
2
answers
166
views
integral solutions of polynomials in two variables
Consider the polynomial
$$
27x^4 - 256 y^3 = k^2,
$$
where $k$ is an integer. As $k$ varies over all positive integers, is it possible to show that there are infinitely many distinct integral ...
- 974
3
votes
1
answer
173
views
Irreducible polynomials with distinct irrational roots and all non-zero coefficients
Is it true that for every natural number $n \geq 3$, there exists an irreducible polynomial $f(x) = \sum_{i=0}^{n} a_{i}x^{i} \in\mathbb{Z}[x],$all of whose coefficients are non-zero and all of its ...
- 416
6
votes
0
answers
81
views
Irreducibility of $(n+1)+nx+(n-1)x^2+\ldots+x^n$ [duplicate]
I'm trying to prove the irreducibility over the rationals of the polynomial defined by
$$f(x)=(n+1)+nx+(n-1)x^2\ldots+x^n$$
for all $n\in\mathbb{N}$. Computationally, I've verified it is always ...
- 89
3
votes
0
answers
110
views
Factorization in $\mathbb{Z}[\sqrt[4]{3}]$ [duplicate]
Let $p$ a prime number such that $p≡3$ (m0d $4$), and $p>3$, discuss the factorization of $(p)$ in $\mathcal{O}_K$, where $K=\mathbb{Q}(\sqrt[4]{3})$.
I know that the ring of integers in this case ...
- 305
1
vote
2
answers
172
views
Polynomials with Unit Discriminant
Let $f=x^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0\in \mathbb{C}[x]$ be a monic polynomial with algebraic integer coefficients and $n>1$. Let $K$ be the number field $\mathbb{Q}(a_{n-1},\dots,a_0)$. ...
11
votes
1
answer
531
views
Can two monic irreducible polynomials over $\mathbb{Z}$, of coprime degrees, have the same splitting field?
Let $f,g \in \mathbb{Z}[X]$ be monic polynomials. It is possible for distinct monic polynomials over $\mathbb{Z}$ to have the same splitting field. For example $f = x^4 - 2$ and $g= x^4+2$ both have ...
- 113
3
votes
3
answers
506
views
Irreducibility of $x^{p(p-1)}+x^{p(p-2)}+\cdots+x^p+1$ over $\mathbb Q$ [duplicate]
It is well known that the $p$th cyclotomic polynomial
$$\Phi_p(x)\ =\ x^{p-1}+\cdots+x+1\ =\ \frac{x^p-1}{x-1}$$
is irreducible over $\mathbb Q$ for prime $p$. The standard trick is to make the ...
- 4,103
2
votes
1
answer
315
views
$X^n + X + 1$ reducible in $\mathbb{F}_2$
I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?
- 415
17
votes
2
answers
4k
views
Minimal polynomial of product, sum, etc., of two algebraic numbers
The standard proof, apparently due to Dedekind, that algebraic numbers form a field is quick and slick; it uses the fact that $[F(\alpha) : F]$ is finite iff $\alpha$ is algebraic, and entirely avoids ...
- 2,560
2
votes
1
answer
88
views
How do we determine the decomposition of $p\mathcal{O}_K$ in $K = \mathbb{Q}(\sqrt[3]{5})$?
Let $K = \mathbb{Q}(\sqrt[3]{5})$, and $\mathcal{O}_K$ be its ring of integers.
In general, how do we decide the decomposition of $p\mathcal{O}_K$, for an odd prime $p$?
I know that by Kummer's ...
- 195
4
votes
1
answer
457
views
Does there exist a finite set of polynomials which do not have roots over any prime field?
The polynomial $x^2 + 1$ has a root in $Z_p$ if and only if $p \not\equiv 3 \mod 4$, and the polynomial $x^2 + x + 1$ has a root in $Z_p$ if and only if $p \not\equiv 2 \mod 3$.
So each of the ...
- 108
4
votes
2
answers
178
views
Polynomial transformation of the roots of another irreducible polynomial.
Suppose I have some monic irreducible polynomial $g(x)$ in $\mathbb{Z}[x]$ with distinct roots $r_1,r_2,\dots,r_n$.
Suppose $f(x)$ is some other polynomial, not necessarily irreducible. Is there ...
- 185
15
votes
4
answers
2k
views
Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?
We know that $f(x)=x^4+1$ is a polynomial irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$.
My question is:
Is it true that if $f(x)$ has a linear factor over $\...
- 15.9k
6
votes
2
answers
500
views
Checking irreducibility of polynomials over number fields
Are there general methods for checking irreducibility of polynomials over number fields? For instance, letting $F = \mathbb{Q}(\sqrt{3})$, I want to know whether $x^3 - 10 + 6\sqrt{3}$ is irreducible ...
- 4,888
3
votes
2
answers
143
views
Conjugates of $12^{1/5}+54^{1/5}-144^{1/5}+648^{1/5}$ over $\mathbb{Q}$
After much manual computation I found the minimal polynomial to be $x^5+330x-4170$, although I would very much like to know if there's a clever way to see this. I suspect there is from seeing that the ...
- 13.1k