Talk:Kummer theory

"Quadratic extension" listed at Redirects for discussion

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User:1234qwer1234qwer4 (talk) 23:33, 1 December 2021 (UTC)

Argument given in the "Recovering $a^{1/n}$..." section is incomplete

The given argument assumes that the element $\alpha$ given below is non-zero.

$\alpha =\sum _{l=0}^{p-1}\zeta _{p}^{l}\sigma ^{l}(\beta )\in K(\beta )$

To do this one needs to choose $\beta$ appropriately by using some version of the linear independence of chracters.

The statement needed is as follows: Given any non-zero tuple $(a_{0},\dots ,a_{p-1})$ of elements in $K$, there exists a $\beta$ in $L$ so that $\sum _{l=0}^{p-1}a_{l}\sigma ^{l}(\beta )$ is non-zero.

This is enough since $\sum _{l=0}^{p-1}\zeta _{p}^{l}=0$. So a $\beta$ which satisfies this independence with $a_{l}=\zeta _{p}^{l}$ does not lie in $K$, so it must be a primitive element.

It is certainly not clear that the required linear holds for all primitive elements $\beta$. kapil (talk) 16:20, 5 December 2025 (UTC)