Timeline for Splitting field for $\mathrm{GL}(2,p)$ - reference request
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 8, 2023 at 8:44 | comment | added | Geoff Robinson | However, a beautiful theorem of Benard and Schacher ( Journal of Algebra, 1972 or so), with an alternate proof by G. Janusz ( Proc AMS, 1972) ) shows that potential odd prime divisors of Schur indices of irreducible characters can be excluded by examination of the character table: if the Schur index of a complex irreducible character $\chi$ of a finite group $G$ is divisible by an integer $m$, then the extension of $\mathbb{Q}$ generated by values of $\chi$ must contain a primitive $m$-th root of unity. | |
| Jun 6, 2023 at 12:04 | history | edited | Mare | CC BY-SA 4.0 |
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| Jun 6, 2023 at 10:54 | comment | added | Dave Benson | To add insult to injury, although the quaternion group has a two dimensional representation over $\mathbb{Q}[\sqrt{-35}]$, it cannot be written as two by two matrices over the ring of integers in this field. | |
| Jun 6, 2023 at 10:35 | comment | added | Dave Benson | @Mare If you want a cautionary example, look at the quaternion group of order eight. All the entries in the character table are integers. But a field $K$ of characteristic zero is a splitting field if and only if $x^2+y^2=-1$ has a solution in $K$. The "generic" one is the field of fractions of $\mathbb{Q}[x,y]/(x^2+y^2+1)$ but there are many smaller ones such as $\mathbb{Q}[\sqrt{-35}]$. | |
| Jun 6, 2023 at 10:14 | comment | added | Benjamin Steinberg | I don’t have access to James’s book but I’ll try to get my hands on it. | |
| Jun 6, 2023 at 9:32 | comment | added | Mare | @DaveBenson Thanks. I try to think about 1. again then. | |
| Jun 6, 2023 at 9:32 | history | edited | Mare | CC BY-SA 4.0 |
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| Jun 6, 2023 at 9:31 | comment | added | Dave Benson | Character tables does not determine splitting fields, because of Schur indices. | |
| Jun 6, 2023 at 8:10 | history | answered | Mare | CC BY-SA 4.0 |