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Along the lines of the general question "How much does the discriminant of a number field reveal about the field?", I was wondering how often it happens that the discriminant of some number field uniquely identifies the Galois group of (the Galois closure over $\mathbb{Q}$ of) that field. Of course not always: there are $S_n$-extensions whose discriminant is the same as that of some quadratic field; on the other hand, certain small discriminants (such as -3, which must belong to the group $C_2$) can only occur once by standard discriminant bounds. Intuitively, I would think that such unique discriminants are pretty common, so:

Q1: Is there a reasonable heuristic for the asymptotic of, e.g., number of degree-$n$ number fields whose discriminant suffices to identify the Galois group? Is it even clear that such fields exist for all $n$?

I was also surprised by the following easy example: If $F/\mathbb{Q}$ is a tamely ramified quartic $A_4$-extension, then its discriminant can NEVER identify the Galois group! Namely, this discriminant would have to be a square; if it involves more than one prime factor, then some $C_2\times C_2$-extension has the same discriminant; but if it has only one prime factor, then this prime must be $\equiv 1$ mod $3$, and so there is a $C_3$-extension with the same discriminant!

Q2: Are there other examples as this, maybe infinite families, of permutation groups $G$ that can never be recognized from the discriminant of a (possibly, tame) $G$-extension?

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  • $\begingroup$ Take $F_1/\mathbb{Q},F_2/\mathbb{Q}$ two (tame) extensions with same discriminant $d$ but different Galois groups $G_1,G_2$. Then for any (tame) Galois extension $L/\mathbb{Q}$ of group $G$ with discriminant $e$ coprime to $d$, the (tame) extensions $LF_1//\mathbb{Q}$ and $LF_2/\mathbb{Q}$ have same discriminant $ed$ but non isomorphic Galois groups $G\times G_1$ and $G\times G_2$ (it is a classical result that these groups are isomorphic iff $G_1\simeq G_2$). Hence you have infinitely many examples of discriminants which cannot determine Galois groups. Did I misunderstand the question ? $\endgroup$
    – GreginGre
    Commented Aug 21, 2018 at 9:51
  • $\begingroup$ @GreginGre: This gives examples of particular discriminant values which are not unique. This happens a lot (for suitable values). What I want in Q2 is examples of groups where there is not even a single discriminant value which is unique. The $A_4$-example is essentially due to a group-theoretical condition (roughly speaking, the group is somehow ``dominated" by the two abelian groups $C_3$ and $C_2\times C_2$), which could a priori work for infinitely many other groups, but I haven't found any other example so far. $\endgroup$ Commented Aug 21, 2018 at 19:02

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