For questions related to cyclotomic polynomials and their properties.

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10
votes
4answers
1k views

showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$

I studied the cyclotomic extension using Fraleigh's text. To prove that Galois group of the $n$th cyclotomic extension has order $\phi(n)$( $\phi$ is the Euler's phi function.), the writer assumed, ...
4
votes
3answers
375 views

Minimal Polynomial of $\zeta+\zeta^{-1}$

Question is to find Minimal Polynomial of $\zeta+\zeta^{-1}\in \mathbb{Q}(\zeta)$ over $\mathbb{Q}$ Where $\zeta$ is primitive $13^{th}$ root of Unity. What all i know is that Minimal polynomial of ...
9
votes
0answers
253 views

What comes after $\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}$?

We have, $$\big(\cos(\tfrac{2\pi}{5})^{1/2}+(-\cos(\tfrac{4\pi}{5}))^{1/2}\big)^2 = \tfrac{1}{2}\left(\tfrac{-1+\sqrt{5}}{2}\right)^3\tag{1}$$ ...
5
votes
1answer
198 views

Cyclotomic Polynomials and GCD

Since Cyclotomic polynomials are irreducible over $\mathbb{Q}$, $\phi_n(x)$, $\phi_m(x)$ are coprime as polynomials in $\mathbb{Z}[x]$. Working over $\mathbb{Q}$, $(\phi_n(x)$, $\phi_m(x))=(1)$. ...
4
votes
2answers
338 views

Determine the degree of the splitting field for $f(x)=x^{15}-1$.

Let $f(x)= x^{15} - 1$. Let $L$ be the splitting field of $f(x)$ over the field $K$. Determine the extension degree $[L:K]$ in each case. a) $K= \Bbb{R}$ b) $K= \Bbb{Q}$ c) $K = ...
11
votes
0answers
252 views

Is $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ an irreducible polynomial over $\mathbb{Q}[X]$?

Let $a, b \in \mathbb{Q}$, with $a\neq b$ and $ab\neq 0$, and $n$ a positive integer. Is the polynomial $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ irreducible over $\mathbb{Q}[X]$? I know that ...
7
votes
2answers
187 views

Explicit Galois theory computation in cyclotomic field

Let $p$ be an odd prime. Let $\zeta=e^{\frac{\pi i}{4 p}}$, thus $\zeta$ is a primitive $8p$-th root of unity. There is a unique $\mathbb Q$-automorphism $\tau$ of the number field ${\mathbb ...
4
votes
2answers
55 views

Irreducibility of $X^n-a$

Let ${\mathbb K}$ be a subfield of ${\mathbb C}$. Let $a\in{\mathbb K}$ such that $X^d-a$ has no root in ${\mathbb K}$, for any divisor $d>1$ of $n$. Does it follow that $X^n-a$ is irreducible ...
6
votes
3answers
108 views

Splitting $\Phi_{15}$ in irreducible factors over $\mathbb{F}_7$

I have to split $\Phi_{15}$ in irreducible factors over the field $\mathbb{F}_7$. It has been a while that I did this kind of stuff, and to be the honest, I've never really understood this matter. I'd ...
2
votes
1answer
169 views

Beating Gauss on Irreducibility of Cyclotomic Polynomials?

I am considering how to provide an alternative proof of the lemma used in the proof that $\Phi_n$ is irreducible: Lemma: If $\Phi_n=f_1 f_2\cdots f_r$ is a factorization into monic irreducible ...