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LSpice
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See this article https://arxiv.org/pdf/math/9812171.pdf . In the second paragraph of Soulé - Perfect forms and the Vandiver conjecture, it is written that: For any natural integer $i ≤ p − 2$$i \le p − 2$, let $C^{(i)}$ be the subgroup of $C$, where the Galois group of $\Bbb Q(\zeta_p)$ over $\Bbb Q$ acts by the $i$-th power of the Teichmuller character.

I know the definition of the Teichmuller character, $\omega(x)=\lim_{n \to \infty} x^{p^n}$, and I can realize it, but I can not understand the quoted paragraph: Galois group acts by the $i$-th power of the Teichmuller character on $H$, what does it mean? I want to realize this.

It is probable that I did not realize the action of the Galois group on the ideal class group, but I think I understand it well enough. So any general explanation on the action of the Galois group on the ideal class group would be welcome too.

See this article https://arxiv.org/pdf/math/9812171.pdf . In the second paragraph, it is written that: For any natural integer $i ≤ p − 2$, let $C^{(i)}$ be the subgroup of $C$, where the Galois group of $\Bbb Q(\zeta_p)$ over $\Bbb Q$ acts by the $i$-th power of the Teichmuller character.

I know the definition of the Teichmuller character, $\omega(x)=\lim_{n \to \infty} x^{p^n}$, and I can realize it, but I can not understand the quoted paragraph: Galois group acts by the $i$-th power of the Teichmuller character on $H$, what does it mean? I want to realize this.

It is probable that I did not realize the action of the Galois group on the ideal class group, but I think I understand it well enough. So any general explanation on the action of the Galois group on the ideal class group would be welcome too.

In the second paragraph of Soulé - Perfect forms and the Vandiver conjecture, it is written that: For any natural integer $i \le p − 2$, let $C^{(i)}$ be the subgroup of $C$, where the Galois group of $\Bbb Q(\zeta_p)$ over $\Bbb Q$ acts by the $i$-th power of the Teichmuller character.

I know the definition of the Teichmuller character, $\omega(x)=\lim_{n \to \infty} x^{p^n}$, and I can realize it, but I can not understand the quoted paragraph: Galois group acts by the $i$-th power of the Teichmuller character on $H$, what does it mean? I want to realize this.

It is probable that I did not realize the action of the Galois group on the ideal class group, but I think I understand it well enough. So any general explanation on the action of the Galois group on the ideal class group would be welcome too.

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Tireless and hardworking
Tireless and hardworking
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Galois group acts by the $i$-th power of the Teichmuller character on $H$

See this article https://arxiv.org/pdf/math/9812171.pdf . In the second paragraph, it is written that: For any natural integer $i ≤ p − 2$, let $C^{(i)}$ be the subgroup of $C$, where the Galois group of $\Bbb Q(\zeta_p)$ over $\Bbb Q$ acts by the $i$-th power of the Teichmuller character.

I know the definition of the Teichmuller character, $\omega(x)=\lim_{n \to \infty} x^{p^n}$, and I can realize it, but I can not understand the quoted paragraph: Galois group acts by the $i$-th power of the Teichmuller character on $H$, what does it mean? I want to realize this.

It is probable that I did not realize the action of the Galois group on the ideal class group, but I think I understand it well enough. So any general explanation on the action of the Galois group on the ideal class group would be welcome too.