Articles of cyclotomic fields

Minimal cyclotomic field containing a given quadratic field?

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 […]

If a cyclotomic integer has (rational) prime norm, is it a prime element?

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 $p$.

what numbers are integrally represented by this quartic polynomial (norm form)

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 […]

Degree of $\mathbb{Q}(\zeta_n)$ over $\mathbb{Q}(\zeta_n+\zeta_{n}^{-1})$

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

Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

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?