Computing Hilbert Class Field of a number field

Question

I'm trying to compute the Hilbert class field of the extension $\mathbb{Q}(\zeta_{5}, \sqrt{-43})$. I know that it has class number 7. I would like to show that it is contained in $\mathbb{Q}(\zeta_{215})$.

Answer 1

The class field of $K = {\mathbb Q}(\zeta_5,\sqrt{-43})$ is not abelian, and in particular not contained in ${\mathbb Q}(\zeta_{43 \cdot 5})$ since its subfields have too much ramification.

The class number $7$ of $K$ is inherited from $k = {\mathbb Q}(\sqrt{-5 \cdot 43})$, which has class number $14$. The quadratic unramified extension of $k$ is $k(\sqrt{5}) \subset K$; the unramified extension of degree $7$ (with Galois group $D_7$ over the rationals by class field theory, where $D_7$ is the dihedral group of order $14$) is given by a root of $$ f(x) = x^7 - 3x^6 + 3x^5 - 5x^4 + 6x^3 - x^2 + 5x - 1 $$ by pari; it also says that the class field of $k$ has class number $1$.

The same root of $f$ generates the class field of $K$.