Constructing quintic number fields with certain splitting behaviour

Question

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 $101833$.

Do there exist infinitely many such fields?

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.

Answer 1

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 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 $(345)$.

Answer 2

I'm not sure what is and isn't written down, but Manjul Bhargava's work on counting quintic fields can be used to get an exact asymptotic for the number of quintic extensions of Q with discriminant a SQUAREFREE integer between 0 and X. In fact, the number of such will be asymptotic to a constant multiple of X. Squarefree discriminant implies your conditions 2 and 3.

This may be massive overkill, for all I know!

Update: Sorry, I didn't pay careful enough attention to DL's condition! No, squarefree doesn't imply condition 3. That's a stronger local condition. But my guess will be that the method Bhargava uses will still give you a positive proportion, though I'm less certain. (Namely: there is still an Euler factor at p given by the total mass of all etale quintic extensions of Q_p satisfying your conditions.)

Answer 3

The point of this answer, building on a comment by "so-called friend Don", is to point out that behavior has density $0$ among all quintic extensions.

My guides here are the introductory portions of The geometric sieve and the density of squarefree values of invariant polynomials and Mass Formulae for Extensions of Local Fields and Conjectures on the Density of Number Field Discriminants, both by Bhargava.

Let $k$ be a field. Recall that an etale $k$-algebra is a direct sum of finitely many finite separable extension fields of $k$. (We'll be in characteristic zero, so you can ignore the adjective separable.) We'll write $\mathrm{Aut}(K/k)$ for the automorphism group of $K$ preserving $k$; this is a finite group. Let $K$ be a degree $p$-adic extension of $\mathbb{Q}_p$ of degree $n$. As I understand the philosophy of these papers, the probability of that the $p$-adic completion of a degree $n$ number field will be isomorphic to $K$ is supposed to be $$\frac{p-1}{p} \frac{1}{\mathrm{Disc}_p(K)} \frac{1}{\# \mathrm{Aut}(K/\mathbb{Q}_p)}.$$

Moreover, these probabilities for distinct primes are supposed to be independent.

For $K$ an etale $\mathbb{Q}_p$ algebra, we write say that $K$ has symbol $(f_1^{e_1}, f_2^{e_2}, \ldots, f_r^{e_r})$ if $K \cong \bigoplus_{i=1}^r K_i$ where $K_i/\mathbb{Q}_p$ is a field extension of ramification degree $e_i$ and residue field extension of degree $f_i$. A lemma in the Mass Formula paper (Prop 2.1) allows us to group together any place where we sum over the set of all etale $\mathbb{Q}_p$-algebras with a given symbol.

So, our desired Euler factor is:

The unramified extensions These are symbols where all the $e_i$ are $1$; namely $(1,1,1,1,1)$, $(2,1,1,1)$, $(3,1,1)$, $(4,1)$, $(5)$. We compute: $$\frac{p-1}{p} \left( \frac{1}{120} + \frac{1}{12} + \frac{1}{8} + \frac{1}{6} + \frac{1}{4} + \frac{1}{5} \right) = \frac{p-1}{p}.$$ It is not a coincidence that the sum came out to $1$; see equation (2.4) in the Mass formula paper.

The other symbols It is also okay to have symbol $(1^2, 1,1,1)$, $(1^2, 2,1)$, $(1^2, 1^2, 1)$ or $(2^2, 1)$. Each of these symbols corresponds to several possible etale $\mathbb{Q}_p$-algebras (for example, for $p$ an odd prime, there are $2$ ramified quadratic extensions of $\mathbb{Q}_p$, so $(1^2, 1,1,1)$ describes two options), but Prop 2.1 in the Mass formula paper means we don't have to think about this. I compute that the respective terms are

$$\frac{p-1}{p} \left( \frac{1}{p} \left( \frac{1}{6} + \frac{1}{2} \right) + \frac{1}{p^2} \left( \frac{1}{2} + \frac{1}{2} \right) \right)$$

Putting everything together, our Euler factor is $$\frac{p-1}{p} \left( 1+\frac{2}{3p} + \frac{1}{p^2} \right) = 1-\frac{1}{3 p} + \frac{1}{3 p^2} - \frac{1}{p^3}.$$

The point is that $$\prod_p \left( 1-\frac{1}{3 p} + \frac{1}{3 p^2} - \frac{1}{p^3} \right) =0.$$ Now, Bhargava's paper doesn't directly allow us to us the prudct in this manner because his best theorem, Theorem 1.3 in the sieve paper, only applies when, for all sufficiently large $p$, we include all the terms with discriminant $1$ or $p$. However, let $p_N$ be the probability that the ramification is as you wish for all $p < N$. Then Bhargava's Theorem 1.3 does show that $p_N$ exists and equals $\prod_{p<N} \left( 1-\frac{1}{3 p} + \frac{1}{3 p^2} - \frac{1}{p^3} \right)$. So this product is an upper bound for the probability that a quintic behaves as desired for all primes. Sending $N \to \infty$, we see that the proportion of quintics with the desired behavior is $0$.