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I have a question about the notation in Silverman's Arithmetic of EC. On page 2, "Notice that the Galois group $G_{\overline{K}/K}$ acts on $\mathbb{A}^n$; for $\sigma\in G_{\overline{K}/K}$ and $P\in \mathbb{A}^n$, $$P^\sigma=(x_1^\sigma,\dots,x_n^\sigma)."$$

Does $x^\sigma=\sigma(x)$?

PatrickStewart
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  • Welcome to MSE. A question should be written in such a way that it can be understood even by someone who did not read its title. – José Carlos Santos Dec 08 '23 at 07:47
  • Yes, I believe $x^\sigma$ means $\sigma(x)$. Is the notation not defined before that? – Greg Martin Dec 08 '23 at 09:29
  • @GregMartin Not that I can see. – PatrickStewart Dec 08 '23 at 09:32
  • An important point: Although $x^{\sigma} = \sigma(x)$, placing the $\sigma$ on the right is significant because it suggests that the author is defining the composition $\sigma \tau$ as do $\sigma$, then $\tau$. This way, $x^{\sigma \tau} = (x^\sigma)^\tau$, which is a much better formula than if the $\sigma$ and $\tau$ were reversed on the right side. – Ted Dec 09 '23 at 02:18

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