All Questions
Tagged with prime-factorization polynomials
6
questions
7
votes
1
answer
403
views
GCD of $n^a\,\prod\limits_{i=1}^k\,\left(n^{b_i}-n\right)$ for $n\in\mathbb{Z}$
Let $a$ be a nonnegative integer. For a given positive integer $k$, let $b_1,b_2,\ldots,b_k$ be odd integers greater than $1$. Using this result, it can be shown that, for each integer $n$, $$f_{a;...
- 49.8k
2
votes
3
answers
104
views
Show factorization $ x^{2n}-1=(x^2-1) \prod_{k=1}^{n-1}(x^2-2x \cos \frac{\pi k}{n} + 1) $
I'm interested in how to show that
$$
x^{2n}-1=(x^2-1) \prod_{k=1}^{n-1}(x^2-2x \cos \frac{\pi k}{n} + 1)
$$
I've seen this equality too often, but have no idea how to derive it. I've tried the ...
- 1,162
2
votes
3
answers
236
views
Number of solutions of polynomials in a field
Consider the polynomial $x^2+x=0$ over $\mathbb Z/n\mathbb Z$
a)Find an n such that the equation has at least 4 solutions
b)Find an n such that the equation has at least 8 solutions
My idea is to ...
- 171
-1
votes
1
answer
596
views
An Analogue of CRT for Polynomial Ring
How can we determine the number of factors by CRT?
For example,
$$\frac{\mathbb{Z}_5[X]}{X^2+1}\cong \frac{\mathbb{Z}_5[X]}{X+2}\times \frac{\mathbb{Z}_5[X]}{X+3},$$
so $\frac{\mathbb{Z}_5[X]}{X^2+...
- 829
2
votes
0
answers
57
views
A question on polynomials.
Let a polynomial $f\in\mathbb{R}[x,y]$, and
$f(x,y)=(x^2+y^2)p(x,y)^2-q(x,y)^2$ and $p,q$ are coprime to each other.
When do, $f$ and $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\...
- 953
0
votes
1
answer
175
views
Prove that the elements $2x$ and $x^2$ have no LCM in the ring of integral polynomials with even coefficient of $x$
Let $A$ be the subring of $\Bbb Z[x]$ consisting of all polynomials with even coefficient of $x$. Prove that the elements $2x$ and $x^2$ have no lowest common multiple.
Hints please!
- 1,332