Timeline for Question on the extension theorem from Humphreys' book on linear algebraic groups
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 29, 2023 at 8:06 | comment | added | Mr Dumas | Oh ok, if $\Lambda$ is an uncountable index set and $R=\oplus_{\lambda\in\Lambda}\mathbb F_p$ and $S$ is the diagonal embedding of $\mathbb F_p$ in $R$ then for any $x=(x_\lambda)\in R$ an equation of integral dependence of $x$ over $S$ is given by the polynomial $t^p-t\in S[t]$. Thanks for the help. | |
| Dec 29, 2023 at 3:56 | comment | added | LSpice | @MrDumas, re, sure; take $S$ to be a finite field and $R$ to be an uncountable direct sum of copies of $S$. (I don't know how many of those restrictions are necessary, but they make it easy to check that it's really an example.) | |
| Dec 29, 2023 at 3:24 | vote | accept | Mr Dumas | ||
| Dec 29, 2023 at 3:16 | comment | added | Mr Dumas | Thank you for your answer, I understand the argument now. So, it is possible to have an integral extension $R/S$ such that for any countable subset $I$ of $R$ that $S[I]$ is a proper subring of $R$? | |
| Dec 28, 2023 at 23:08 | history | answered | LSpice | CC BY-SA 4.0 |