The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows from the paper "Number fields unramified away from $2$" by John W. Jones in the Journal of Number Theory in 2012.

The existence follows from the fact that $\mathbb{Q}(\zeta_{64})$ has class number $17$ and hence if $H$ is the Hilbert class field of $\mathbb{Q}(\zeta_{64})$, then $H/\mathbb{Q}$ is Galois with $G = Gal(H/\mathbb{Q})$ of order $32 \cdot 17$. If $P$ is a Sylow $2$-subgroup of $G$, then the fixed field of $P$ is a number field $K/\mathbb{Q}$ of degree $17$ ramified only at $2$. (Actually, one of the degree $16$ subfields of $\mathbb{Q}(\zeta_{64})$ also has class number $17$, and the Galois closure of $K/\mathbb{Q}$ has degree $272$. Harbater shows that this is the unique Galois extension of $\mathbb{Q}$ of degree $272$ ramified only at $2$.) My question is the following:

Is it possible to explicitly compute a degree $17$ polynomial $f(x)$ that has a root in $K$?

The motivation for this question is the following. If $C/\mathbb{Q}$ is a curve of genus $g$ with good reduction away from $2$ and $J(C)$ is the Jacobian of $C$, then $\mathbb{Q}(J(C)[2^{n}])/\mathbb{Q}$ is a Galois extension, ramified only at $2$, with Galois group contained in a subgroup of $GSp_{2g}(\mathbb{Z}/2^{n} \mathbb{Z})$. Knowing something about the number fields that can arise allows one to understand the arithmetic of $C$ (and in particular, allows one to find etale covers of degree $2^{n}$). For example, since there are no $A_{3}$ or $S_{3}$ extensions of $\mathbb{Q}$ ramified only at $2$, every elliptic curve $E/\mathbb{Q}$ with good reduction away from $2$ must have a rational $2$-torsion point. (Of course, there are other ways to see this.)

Here are three possible approaches to the question:

$\bullet$ Use existing software that will do computations with class field theory. As a test case, I tried using Magma to compute Hilbert class fields of cubic fields with class group of prime order $p$. For $p \leq 11$, the computation was doable, but for $p > 11$, it seemed quite difficult. Maybe there are more sensible ways to attempt the computation.

$\bullet$ I've seen hints in the literature at generalizations of explicit class field theory for imaginary quadratic fields to CM fields. Maybe there's a way to construct $K$ using CM abelian varieties.

$\bullet$ The rigidity method (sometimes) allows one to construct a Galois extension $L/\mathbb{Q}(t)$ with a given Galois group, and has the ability to construct specializations with a small number of ramified primes. This method seems promising, although $\mathbb{Z}/17\mathbb{Z} \rtimes (\mathbb{Z}/17/\mathbb{Z})^{\times}$ does not have a rationally rigid triple of conjugacy classes.

  • 1
    $\begingroup$ Lassina Dembele has constructed two curves (one of genus $16$ and one of genus $40$) whose $2$-torsion field is $H$. See the preprint here. $\endgroup$ – Jeremy Rouse Nov 15 '17 at 14:21

Thank you for calling this problem to my attention. I computed $K$ en route to AWS (though this year's topics are a rather different flavor of number theory...). After some simplification (gp's $\rm polredabs$), it turns out that the field $K$ is generated by a root of $$ f(x) = x^{17} - 2x^{16} + 8x^{13} + 16x^{12} - 16x^{11} + 64x^9 - 32x^8 - 80x^7 $$ $$ \qquad\qquad\qquad {} + 32x^6 + 40x^5 + 80x^4 + 16x^3 - 128x^2 - 2x + 68. $$ gp quickly confirms:

p = Pol([1,-2,0,0,8,16,-16,0,64,-32,-80,32,40,80,16,-128,-2,68])
F = nfinit(p);


[2 79]

So the field has discriminant $2^{79}$.

Let $F = {\bf Q}(\zeta_{64}^{\phantom{0}} - \zeta_{64}^{-1})$, with $F/\bf Q$ cyclic of degree $16$ of $\bf Q$. It was known that $K$ is contained in an unramified cyclic extension $L/F$ of degree $17$. Hence $L(\zeta_{17})$ is a Kummer extension of $F_{17} := F(\zeta_{17})$. Now $F_{17}$ is a $({\bf Z} / 16 {\bf Z})^2$ extension of $\bf Q$. Fortunately it was not necessary to work in the unit group of this degree $256$ field: the Kummer extension is $F_{17}(u^{1/17})$ with $u \in F_{17}^* / (F_{17}^*)^{17}$ in an eigenspace for the action of ${\rm Gal}(F_{17}/{\bf Q})$, which makes it fixed by some index-$16$ subgroup of this Galois group. Thus $u$ could be found in one of the cyclic degree-$16$ extension of $\bf Q$ intermediate between $\bf Q$ and $F_{17}$, and eventually a generator of $K$ turned up whose minimal polynomial, though large, was tractable enough for gp's number-field routines to do the rest.

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