Timeline for Recognizing the Galois group from the field discriminant
Current License: CC BY-SA 4.0
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| Aug 21, 2018 at 19:02 | comment | added | Joachim König | @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. | |
| Aug 21, 2018 at 9:51 | comment | added | GreginGre | 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 ? | |
| Aug 14, 2018 at 2:21 | history | edited | Joachim König | CC BY-SA 4.0 |
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| Aug 13, 2018 at 17:15 | review | First posts | |||
| Aug 13, 2018 at 18:26 | |||||
| Aug 13, 2018 at 17:10 | history | asked | Joachim König | CC BY-SA 4.0 |