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Let $L$ be a finite abelian extension of $\mathbb{Q}$ and let $m$ be a positive integer such that $L\subset\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive $m$-th root of unity. Let $a$ be an integer coprime to $m$. Then the Artin symbol $(\frac{L}{a})$ is the automorphism of $L$ obtained by restricting to $L$ the automorphism $\phi$ of $\mathbb{Q}(\zeta)$ determined by $(\zeta\mapsto\zeta^a)$.

My question is, why is, $\phi(L)\subset L$?

  • what are your thoughts? – mathworker21 Dec 27 '18 at 12:31
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    Hint: what does it mean for an extension to be abelian? – Wojowu Dec 27 '18 at 12:34
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    @Wojowu: Thanks for the hint. $\phi$ restricts to a $\mathbb{Q}$-homomorphism of $L\to\phi(L)$. Since $L/\mathbb{Q}$ is Galois, $\phi(L)\subset L$. Is this correct? Also, it seems to me that $\mathbb{Q}(\zeta)/\mathbb{Q}$ is not always a cyclic Galois extension, right? –  Dec 27 '18 at 12:51
  • This is correct. Cyclotomic extensions are always Galois (note the conjugates of $\zeta$ are powers of $\zeta$) – Wojowu Dec 27 '18 at 12:53
  • Also, I think the Artin symbol $(\frac{L}{a})$ depends not only on $a$, but also on $m$, am I right? –  Dec 27 '18 at 13:00
  • That is a good follow up question, sai (+1). Initially one might think that it also depends on $m$, but actually it doesn't in the following sense. If we view some other cyclotomic field containing $L$, say $\Bbb{Q}(\xi)$ where $\xi$ is a primitive root of unity of order $\ell$, where $m\mid \ell$ so that $\Bbb{Q}(\zeta)\subseteq \Bbb{Q}(\xi)$. Then the automorphism of $\Bbb{Q}(\xi)$ that maps $\xi\mapsto\xi^a$ is an extension of the automorphism $\zeta\mapsto\zeta^a$. – Jyrki Lahtonen Dec 27 '18 at 13:10
  • There is scope for some ambguity in the sense that some other automorphism $\xi\mapsto \xi^b$, $b\neq a$, may also extend $\zeta\mapsto\zeta^a$. But, obviously both of those automorphisms have the same restriction to $L$, so this doesn't really matter. – Jyrki Lahtonen Dec 27 '18 at 13:11
  • The upshot is that the action on roots of unity gives a sort of uniform way of putting something like a coordinate on the abelian Galois groups over $\Bbb{Q}$. – Jyrki Lahtonen Dec 27 '18 at 13:13
  • @Jyrki Lahtonen: Thank you for the explanations. –  Dec 28 '18 at 09:32

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