There was an exercise labeled difficile (English: difficult) in the material without solution: Suppose $d\in\mathbb Z\backslash\{0,1\}$ without square factors, and $n$ is the smallest natural number $n$ such that $\sqrt d\in\mathbb Q(\zeta_n)$, where $\zeta_n=\exp(2i\pi/n)$. Show that $n=\lvert d\rvert$ if $d\equiv1\pmod4$ and $n=4\lvert d\rvert$ if $d\not\equiv1\pmod4$. It’s easier to show that $\sqrt d\in\mathbb Q(\zeta_n)$, although I […]

Let $p$ be a rational prime. Consider the ring of integers $\mathbb{Z}[\zeta_p] $ of the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$. If the norm $N(\alpha)$ of $\alpha \in \mathbb{Z}[\zeta_p]$ is a rational prime, must $\alpha$ be a prime element of $\mathbb{Z}[\zeta_p] $? If it helps, I only need the case where $N(\alpha) \equiv 1$ mod $p$.

My last question on this was a bit of a dud. I would, at least, like confirmation that the polynomial $f$ below does, in fact, integrally represent all primes $p \equiv 1 \pmod 5,$ along with all squared primes $q^2$ when prime $q \equiv 4 \pmod 5.$ It is easy to show that the polynomial […]

Let $\zeta_n$ be a $n$-th primitive root of unity. How to prove that $[\mathbb{Q}(\zeta_n):\mathbb{Q}(\zeta_n+\zeta_{n}^{-1})]=2$ ?

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?

Intereting Posts

Compute $ \sum_{k=1}^{\infty} \text{sech}(2 k)$
Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$
Does every nonempty definable finite set have a definable member?
What is a Manifold?
proving $ \sqrt 2 + \sqrt 3 $ is irrational
Fermat's Last Theorem simple proof
Is there a vector space that cannot be an inner product space?
How many different sets of $6$ different numbers can we construct from $11, 13, 18, 19, 19, 20, 23, 25$?
How can I Prove that H=G or K=G
When is $\mathrm{Hom}(A,R) \otimes B =\mathrm{Hom}(A,B)$?
Elementary solution to the Mordell equation $y^2=x^3+9$?
If $f\tau$ is continuous for every path $\tau$ in $X$, is $f:X\rightarrow Y$ continuous?
Continuous bijection from $(0,1)$ to $$
If $gHg^{-1} \subset H$, must we have $g^{-1}Hg \subset H$?
Number of unique cubes with one red cube in every $1*1*4$ segment