Timeline for How to compute Hilbert class field of $\Bbb Q(\zeta_{31})$?
Current License: CC BY-SA 4.0
10 events
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| Jan 3, 2021 at 13:06 | comment | added | Alex B. | @SunilPasupulati: Magma told me so (I did the computation conditional on GRH). However, the argument does not really depend on that. If the group was $C_3\times C_3$, the entire argument would still work: the extension would be split, and the subgroup of order $5$ in $I$ would still act trivially, so that you could find the Hilbert class field by looking at the sextic subfield of $\mathbb{Q}(\zeta_{31})$. | |
| Jan 3, 2021 at 5:21 | comment | added | SUNIL PASUPULATI | @AlexB. How we say the class group of K is cyclic? | |
| Jan 2, 2021 at 20:30 | history | edited | Alex B. | CC BY-SA 4.0 |
Fixed some typos, mentioned a nicer polynomial that generates the Hilbert class field, following a comment by Franz Lemmermeyer
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| Jan 2, 2021 at 19:02 | comment | added | Franz Lemmermeyer | I think I just searched the number field data base - it's been a while since I did that, so I don't remember the details. I guess you could also search for all cubic extensions of the cubic field with discriminant $-31$ satisfying the obvious conditions - if there's only one such field then you're done. I gave a list of Hilbert class fields of cyclotomic fields in my foreword to the English translation of Hasse's book on class numbers of abelian number fields mathi.uni-heidelberg.de/~flemmermeyer/publ/Hpr2.pdf | |
| Jan 2, 2021 at 18:45 | comment | added | Alex B. | @FranzLemmermeyer: it is! How did you find that polynomial? Do feel free to edit the answer, or alternatively I could edit in your much nicer polynomial, but I would be very interested in how you arrived at it. | |
| Jan 2, 2021 at 17:23 | comment | added | Franz Lemmermeyer | Can Magma verify that this is the same field as the one you get by adjoining a root of $x^9 − x^7 − 2x^6 + 3x^5 + x^4 + 2x^3 − x^2 + x − 3$? | |
| Jan 2, 2021 at 11:50 | comment | added | Alex B. | @FrançoisBrunault: thank you! I did not try to reduce the equation. If someone does, they should feel free to edit the answer. | |
| Jan 2, 2021 at 7:26 | comment | added | François Brunault | That's impressive. Did you try to reduce the defining equation? One could use 'rnfpolredabs' or 'rnfpolredbest' in Pari/GP (there are probably similar commands in Sage or Magma). | |
| Jan 1, 2021 at 15:50 | history | edited | Alex B. | CC BY-SA 4.0 |
added 884 characters in body
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| Jan 1, 2021 at 15:14 | history | answered | Alex B. | CC BY-SA 4.0 |