For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \{1 \text{ for } n=1,2; (1/2)\phi(n) \text{ for } n>2\}$. $\phi$ is the Euler totient function which gives the number of coprime elements. We are new to abstract algebra. This is a question on a difficult project involving cyclotomic polynomials and their irreducibility. We know that the degree of […]

This is a continuation of the question Factorization of polynomial with prime coefficients from earlier today. As in the linked question we are interested in the possibility of a polynomial of the form $$ f(x)=p_0+p_1x+\cdots+p_{n-1}x^{n-1} $$ having a common factor with $x^n-1$ in the ring $\Bbb{Z}[x]$. Here the coefficients $p_i$ are constrained to be distinct […]

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 ${\mathbb K}={\mathbb Q}(\zeta)$ such that $\tau(\zeta)=\zeta^{4p-1}$. Consider the number $$ x=\prod_{k=1}^{p} \tan\big(\frac{\pi k}{4 p}\big)=i^p \prod_{k=1}^{p} \frac{\zeta^k+\zeta^{-k}}{\zeta^k-\zeta^{-k}} $$ Can anybody show that $x$ is fixed by […]

Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as $x=f(-2)$, but I don’t know how to find it “by hand”. This led me to think about $\sqrt{2}$ roots of unity […]

If $p \in \mathbb{N}$ is a prime, is $\displaystyle A=\frac{x^{p^{2}}-1}{x^{p}-1}$ irreducible in $\mathbb{Q}[x]$? I don’t think it is. If somebody sees a contradiction, I would be glad to see it. The link of Qiaochu links to the Eisenstein criterion. There it is written that: cyclotomic polynomials can be obtained by dividing the polynomial $x^{p}-1$ (in […]

I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible $$ \frac{x^p – 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$ He shows this polynomial is irreducible in $\mathbb{F}_q[x]$ whenever $p$ is a primitive root mod $q$. By Dirichlet’s theorem there are infinitely many primes […]

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The irreducibility of its other factor, $$\dfrac{x^{6k+2}-x+1}{x^2-x+1}$$ holds for all k lesser than $790$, at the very least. My question would be whether it holds $~\forall~k\in\mathbb N$. […]

This question already has an answer here: Irreducible cyclotomic polynomial 1 answer

Let $\zeta\in \mathbb C$ be a primitive $7^{th}$ root of unity. Show that there exists a $\sigma\in \operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ such that $\sigma(\zeta)=\zeta^3$. I already know that $\zeta$ is a root of $f(x)=x^6+x^5+x^4+x^3+x^2+x+1$ and that $f$ is irreducible (By applying Eisenstein’s criterion on $f(x+1)$). Also the powers of $\zeta$ are also roots of $f$. So […]

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 $\bigl(X(X-a)\bigr)^{2^n} +1$ is irreducible over $\mathbb{Q}[X]$, but I have a hard time generalizing my proof with three factors. PS: This is not homework (and may even be open).

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