(This is related to one of my previous questions; reading is not required though)
I'm still banging my head on the following exercise.
Consider the primitive $n$-th root of unity $\zeta_{n} := exp(\frac{2\pi i}{n})$. Show that the number field $K := \mathbb{Q}(\zeta_{20})$ has class number one.
In the exercise, the following hint is given:
Show that it suffices to show that any prime ideal above the primes $2,3,5,7,11$ is principal. We know that the quadratic subfields of $\mathbb{Q}(\zeta_{20})$ are $\mathbb{Q}(\sqrt{-5})$, $\mathbb{Q}(\sqrt{5})$, $\mathbb{Q}(i)$. The prime 2 may be treated via $\mathbb{Q}(i)$. For 3 and 7, observe that $\omega_1^2 + \omega_2^2 = 3$ and $\omega_1^4 + \omega_2^4 = 7$, where $\omega_1 := (1+\sqrt{5})/2$ and $\omega_2 := (1-\sqrt{5})/2$. For 5 show that the norm from $\mathbb{Q}(\zeta_{20})$ to $\mathbb{Q}(\zeta_{5})$ of $(\zeta_5 + \zeta_5^{-1})+\zeta_5^2\cdot i$ is $1-\zeta_5$. For 11, first determine its prime factors in $\mathbb{Q}(\zeta_5)$.
What I have accomplished so far:
I have shown via Minkowski-bound that it suffices to show that any prime ideal above the primes $2,3,5,7,11$ is principal.
I also managed to show that any prime ideal above $2$ or $11$ must be principal.
So it remains to show that for the primes $3,5,7$, obviously using the hints above. However I don't really seem to get to the point where the hints make sense to me.
If it helps, I also computed the inertia/decomposition fields: Given a prime number $p$, let $r$ denote number of prime ideals of $\mathbb{Z}[\zeta_{20}]$ above $p$. Then let $e$ be ramification index and $f$ the inertia degree. For primes $3,5,7$ get:
- Prime 3: $r=2, e=1, f=4$. Decomposition field = $\mathbb{Q}(\sqrt{-5})$, inertia field = $K$
- Prime 5: $r=2, e=4, f=1$. Decomposition field = inertia field = $\mathbb{Q}(i)$
- Prime 7: $r=2, e=1, f=1$. Decomposition field = $\mathbb{Q}(\sqrt{-5})$, inertia field = $K$
Thanks for any help in advance, I'd really like to close this chapter.