All Questions
Tagged with cyclotomic-fields polynomials
8 questions
3
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0
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135
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Recover cyclotomic integer with bounded coefficients from its known associate
Recall that two cyclotomic integers are called associated if their ratio is a unit in the corresponding ring of cyclotomic integers.
We will view cyclotomic integers as polynomials (of degree $<\...
- 34.3k
3
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Real root of the derivative of a prime cyclotomic polynomial
Consider the graph of $y=x^n$ with $n$ odd, and draw a tangent to its negative arc that crosses the graph at $(1,1)$. Equating the slopes gives us $\frac{x^n-1}{x-1}=nx^{n-1}$ at the point of tangency....
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1
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1
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How to find a cyclotomic polynomial of degree d that decompose into d irreducible polynomials in $Z_6$?
More specifically, I need the degree $d$ to be around 1024. I can easily find the cyclotomic polynomial of degree 1024 that satisfies the above requirement in $Z_2$, i.e., $x^{1024}+1$, which is equal ...
- 156k
13
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1
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Products of Cyclotomic Polynomials with Nonnegative Coefficients
I'm curious if there are any results that allow us to determine if a product of cyclotomic polynomials (not necessarily all distinct) results in a polynomial having nonnegative coefficients.
Some ...
- 30.1k
1
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On largest degree of polynomial related to cyclotomic polynomials - I
We know cyclotomic polynomials $\Phi_{2^kp^rq^m}(x)$ have coefficients in $\{0,\pm1\}$.
What is the largest degree $f_{d,n}(x)=\frac{x^{2^{k}p^{r}q^{m}}-1}{\Phi_d(x)}$ with $\{0,\pm1\}$ coefficients ...
- 13.9k
5
votes
1
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Products of cyclotomic polynomials
Is $\Phi_5(z) \Phi_6(z) = 1 + z^2 + z^3 + z^4 + z^6$ the only product of cyclotomic polynomials that has nonnegative coefficients and satisfies $p(\zeta)=0$, $p(\zeta^2)=2$, $p(\zeta^3)=3$, and $p(1)=...
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- 1
11
votes
0
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Evaluating products of cyclotomic polynomials at roots of unity
Are there general non-trivial conditions on $p(\cdot)$ and $n$, where $p(\cdot)$ is a product of cyclotomic polynomials and $n$ is a positive integer, such that all the coefficients of $p(\cdot)$ are ...
- 19.7k
6
votes
1
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Which criteria for "uniformly splitting" polynomials?
Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials ...
- 20.4k