# Tagged Questions

For questions related to cyclotomic polynomials and their properties.

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### Series of Cyclotomic polynomials

How can I show that if $\Phi$ is a Cyclotomic polynomial, $$\Phi_n(x)=\prod_{1\leq k\leq n}_{(n,k)=1}(x-\zeta_n^k)$$ With $\frac{d}{dx}\Phi_n(x)=\Phi'_n(x)$ Then, ...
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### How many non-rational complex numbers $x$ have the property that $x^n$ and $(x+1)^n$ are rational?

In this answer, I proved the following statement: For each positive integer $n$, there are finitely many non-rational complex numbers $x$ such that $x^n$ and $(x+1)^n$ are rational. These complex ...
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### Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ...
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### Derivatives of the nth cyclotomic polynomial

Are there any useful properties of the $k$th derivative of the $n$th cyclotomic polynomial? In particular, what would the value of this be at $1$ and $0$, or any properties of the value of the $k$th ...
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### Gaussian periods

Let p be an odd prime number and $p-1=m d$ a decomposition into positive factors. Then there is a unique cyclic extension $K_d/\mathbb Q$ of degree d ramified only at p. It is contained in the ...
### Is is true that $\zeta$ has finite order?
Let $\zeta$ be a complex number on the unit circle $\{z\in \mathbb{C}: |z|=1\}$.Suppose that $[\mathbb{Q}(\zeta):\mathbb{Q}] < \infty$.Is it true that $\zeta ^n=1$ for some positive integer $n$?
### Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$
Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$ I have asked a similar problem Minimal Polynomial of $\zeta+\zeta^{-1}$ and i tried to repeat similar idea ...