Imaginary Non-CM Fields of $2$-Power Degree

Source abstract

We establish a new method for studying imaginary non-CM fields of arbitrary $2$-power degree. In particular, we show that for degrees greater than $16$, dihedral fields either lie in the ray class field of one of the three imaginary quadratic fields $\mathbb{Q}(\sqrt{-2})$, $\mathbb{Q}(\sqrt{-3})$, $\mathbb{Q}(\sqrt{-67})$, with the conductor of this ray class field supported only on primes dividing $2$ and with explicit upper bounds on the relevant exponents, or are Hilbert class fields of imaginary quadratic fields. We also give a complete classification of the $104$ imaginary non-CM fields of degree $16$ with class number one, providing explicit tables of defining polynomials, Galois groups, base fields, and conductors. These 104 non-CM fields, together with the five CM fields classified by Louboutin and Okazaki \cite{lou7}, complete the classification of all imaginary fields of degree 16 with class number one. For CM fields, the problem reduces to computing relative class numbers via analytic class number formulae, whereas for non-CM fields the absence of a totally real subfield of index $2$ necessitates a further approach and leads to a richer variety of Galois groups and ramification patterns.