Families of number fields of prime discriminant

Question

When I am testing conjectures I have about number fields, I usually want to control the ramification, especially minimize to a single prime with tame ramification. Hence, I usually look for fields of prime discriminant (sometimes positive, sometimes negative).

I get the feeling that I cannot be the only one who does this...

And so, are there families of number fields of prime discriminant for each degree? Or at least degree 3 and 4? (They are the coolest. Except quadratics. Of course.) What about: given a prime - can I find a polynomial of degree d with the prime as its discriminant?

Answer 1

Klueners Malle online might be just the thing you're looking for. Make your own lists! And here's some they made themselves, if you run out of ideas.

Answer 2

A recent paper of Bhargava and Ghate discusses the enumeration of quartic fields of prime discriminant (see section 7).

Answer 3

As for your second question: if $f$ is an irreducible polynomial woth degree $n$ and prime discriminant (actually, squarefree discriminant is sufficient), then the roots of $f$ generate an extension with Galois group $S_n$ (thus these examples are not necessarily the best candidates for testing conjectures since you miss out on all the more interesting Galois groups). Since $S_n$ is not solvable for $n \ge 5$, class field theory is your friend only for $n = 2, 3, 4$.

• For $n = 2$, the situation is trivial.
• For $n = 3$, there is a number field with prime discriminant $p$ if and only if the
  quadratic field with discriminant $p$ has class number divisible by $3$.
• For $n = 4$, there is an $S_4$-extension with prime discriminant if and only if there is a quadratic number field with discriminant $p$ and class number divisible by $3$ such that one of its unramified cubic extensions has class number divisible by $2$ (and thus necessarily by $4$).

There's a very nice article by Shanks (A survey of quadratic, cubic and quartic algebraic number fields (from a computational point of view), Proc. 7th southeast. Conf. Comb., Graph Theory, Comput.; Baton Rouge 1976, 15-40 (1976)) where you will find more.

Scholz, during the 1930s, showed how to construct (using class field theory, which means you will not get generators, just the existence) of Galois groups with small solvable groups; in his construction, the number of ramified primes can be controlled.

Answer 4

John Jones' Tables http://hobbes.la.asu.edu/NFDB/ are my favorite. I think his data tables are the most complete online. Check for example in Klueners-Male tables for cubic fields of prime discriminant -3299, and you'll see that there are no results shown. However Jones' tables contains the 4 cubic fields with such discriminant.

Now about your question on the primes, as Ben mentioned p must be 1 mod 4 so the question has some hope. Even for p that are "allow" to be discriminants there might not exist a field of fix degree d of discriminant p. For example, class field theory tells you that there is a cubic field of discriminant p if and only if the 3-Sylow part of $Cl(\mathbb{Q}(\sqrt{p}))$ is non-trivial. In fact you can even tell how many of them there are! As an example of this we can conclude that there are no cubic fields of discriminant p=-3, even though p is a fundamental discriminant. A similar analysis can be done for quartic fields, and I think for them the behavior is related to the 2-Sylow part of $Cl(\mathbb{Q}(\sqrt{p}))$.

Also you might want to look at this paper of Jone's which I think is very close to your question http://hobbes.la.asu.edu/papers/OnePrimeJR.pdf

Answer 5

I am going to annoy all the people who answered above, but I am pretty sure the answer to Dror's question is basically no. In particular, is it known that there are infinitely many cubic fields of prime discriminant? I have not heard of such a result -- if one is out there then I would be extremely grateful if someone would share the appropriate references with me.

As is pointed out above, there is a classical correspondence between such fields and subgroups of $Cl(\mathbb{Q}(\sqrt{p}))$ of index 3. However, I'm not aware that this makes the question easier to answer.

There is also the work of Bhargava and Ghate, Delone-Faddeev, Davenport-Heilbronn, etc. which says that cubic (and quartic and quintic) fields are parameterized by integral orbits on nice prehomogeneous vector spaces which meet certain local conditions. For example, in the cubic case, cubic rings are parameterized by integral binary cubic forms up to $GL_2(\mathbb{Z})$ equivalence, and maximal cubic orders are those cubic rings which meet a certain local condition at each prime.

This allows you to prove formulas for the number of cubic fields with $Disc(K) < X$ with good error terms, and this works if you ask for the condition $d | Disc(K)$. This allows you to run a sieve.

However sieves are notoriously bad at finding primes! The information above is also essentially available in the twin prime problem, but all we can prove there is that there are infinitely many primes $p$ so that $p + 2$ has at most two prime factors.

You can use this argument to find cubic fields with three (I think) prime factors -- there is a paper of Belabas and Fouvry that does this. Maybe you could push their arguments a little bit better. But one cannot hope to find primes this way.

Of course there are excellent computational results, and I don't want to take anything away from these. But I feel like the question is asking if there are infinite families, and I'm pretty sure this is widely expected but not at all known.