# Questions tagged [cyclotomic-polynomials]

For questions related to cyclotomic polynomials and their properties.

313 questions
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### Let $n>2$ prove that $[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$

Let $n>2$ prove that $\text{deg}(m_{z+1/z}):=[z+\frac{1}{z}:\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$ So, since I know that with $z$ being a primitive $n$-th ...
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### How to show that a given polynomial is irreducible in a cyclotomic field

I'm beginning to study McCarthy's Algebraic Extensions of Fields, and one of the first problems is to give a factorization of $x^4 + 1$ in $K[x]$, where $K=\mathbb{Q}(a)$ and $a$ is a root of $x^4+1$ (...
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### Quadratic polynomial satisfied by $\zeta_5+\zeta_5^{-1}$

I got one problem from Dummit Foote stating that determine the quadratic polynomial satisfied by the period $\alpha=\zeta_5+\zeta_5^{-1}$ of the the $5th$ root of unity $\zeta_5$. Determine the ...
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### Does there exist a formula for product of the primitive $n$th roots of unity.

I know there is a formula for the sum of the primitive $n$th roots of unity which is the Mobius function of $n$. See: The Möbius function is the sum of the primitive $n$th roots of unity. I ...
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### How to program formula involving cyclotomic polynomials and Lambert series?

I want to know how to program this formula (https://people.math.gatech.edu/~mschmidt34/images/sum-of-divisors-exact-formula.png) but I can't understand the math behind or the several variables used ...
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### Show the Galois group of $f(x)=x^p - 2$ is isomorphic to $\mathbb{Z}_p$

I'm having some difficulty with a problem. Here is the problem statement. Let $p$ be a prime, and $G$ be the Galois group of $f(x) = x^p - 2$ over $E$ where $E = \mathbb{Q}(\xi)$ and $\xi$ a ...
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### Let $n \geq 3$ and let $p$ be prime. Show that $\sqrt[n]{p}$ is not contained in a cyclotomic extension of $\mathbb{Q}$ [duplicate]

I think that the following statement is correct. Let $n \geq 3$ and let $p$ be prime. Show that $\sqrt[n]{p}$ is not contained in a cyclotomic extension of $\mathbb{Q}$. My idea for a proof is ...
### Proving the identity $\Phi_{np}(x) = \Phi_n(x^p)/\Phi_n(x)$, with $p \nmid n$
I've been going through the Wikipedia article on cyclotomic polynomials, trying to prove the various identities that can be used as tools to compute $\Phi_n(x)$. One such identity that I'm not sure ...