I am looking for number fields $K$ which satisfy the following properties:


1. $[K:\mathbb{Q}]=5$.
2. The Galois closure of $K$ has Galois group $S\_5$.
3. For each prime $p$ which *ramifies* in $K$, there exists a prime ideal $\mathfrak{p}$ of $K$ of inertia degree and ramification index $1$ above $p$, i.e., we have
$$(p) = \mathfrak{p} \cdot \prod\_{i=1}^r \mathfrak{p}\_i^{e\_i},$$
with $N(\mathfrak{p}) = p$.


It is well-known that one can construct infinitely many fields satisfying conditions 1 and 2 using Hilbert's irreducibility theorem. It is the last condition 3
which is the most important one, and says something like the ramification of $K$ is very mild.


I have been able to find such fields by looking at databases of number fields. For example, one such field is given by
$$t^5-t^4-5t^3+5t^2+2t-1 =0.$$
This has discriminant equal to $101833$, which is prime. One checks that we have the factorization
$$(101833) = \mathfrak{p}\_1^2 \mathfrak{p}\_2\mathfrak{p}\_3\mathfrak{p}\_4,$$
where each ideal has norm $101833$.



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I'm possibly willing to weaken condition 2 to ask instead that the Galois group is a solvable (but still non-abelian) subgroup of $S\_5$, if it helps. However, in my application I cannot remove condition 1.



Constructing quintic number fields with certain splitting behaviour
Since you are okay with other non-abelian Galois groups, let $F$ be a quadratic extension of $\mathbb{Q}$ with class number divisible by $5$. (There are infinitely many, see [this paper](http://www.math.snu.ac.kr/~dhbyeon/23_class57-2.pdf) and references therein). By class field theory, $F$ has an unramified $\mathbb{Z}/5$ extension $L$. I claim that $L/\mathbb{Q}$ is a $D\_5$ extension, and I claim that the degree $5$ subextensions corresponding to the order $2$ subgroups of $D\_5$ have the desired property.


**Verification that $L/\mathbb{Q}$ is a $D\_5$ extension** Let $H$ be the class group of $F$. Let $A/L$ be the unramified extension corresponding by class field theory to $H/5H$. Then $A/\mathbb{Q}$ is Galois and we have a short exact sequence $0 \to H/5H \to \mathrm{Gal}(A/\mathbb{Q}) \to \mathbb{Z}/2 \to 0$. We know that $\mathrm{Gal}(L/\mathbb{Q})$ acts by negation on $H$, and $GCD(|H/5H|,2)=1$, so the short exact sequence is semidirect, and $\mathrm{Gal}(A/\mathbb{Q}) \cong \mathbb{Z}/2 \ltimes H / 5H$. In particular, if we take a particular $\mathbb{Z}/5$ quotient of $H/5 H$, we get a $D\_5$ extension of $\mathbb{Q}$.


**Verification of condition (3)** Let $p$ be a prime which ramifies in $K$ (and hence in $L$) and choose a prime $\pi$ of $L$ lying above $p$. Since $L/F$ is unramified, the ramification degree of $\pi/p$ must be $2$. Let $I\_{\pi}$ be the inertia group and $D\_{\pi}$ the decomposition group. Now, $I\_{\pi}$ is $2$, so it is conjugate to $\{\pm 1 \} \ltimes \{ 0 \}$ in $D\_5$. Since $I\_{\pi}$ is normal in $D\_{\pi}$, we must have $D\_{\pi} = I\_{\pi}$. 


But then we compute that $p$ factors in $K$ as $\mathfrak{p}\_1^2 \mathfrak{p}\_2^2 \mathfrak{p}\_3$, and $\mathfrak{p}\_3$ is the desired prime.


The problem in $S\_5$ is that we can have $I\_{\pi}$ be the group generated by $(12)$ and $D\_{\pi}$ the group generated by $(12)$ and $(345)$.


