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.



Degree 17 number fields ramified only at 2
Thank you for calling this problem to my attention.
I computed $K$ en route to AWS (though this year's
[topics](http://swc.math.arizona.edu/index.html)
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])
poldegree(p)
F = nfinit(p);
factor(F.disc)

```

returns



```
[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.


