I'm reading Washington's book on cyclotomic fields, and he mentions that it is sometimes convenient to embed $\mathbb{C}_p$ into $\mathbb C$ and vice versa. In my mind, $\mathbb C_p$ and $\mathbb C$ seem like fundamentally different objects, so I'm wondering what such embeddings would look like, or how to think about this. In particular, when speaking of things like the Teichmuller character $\omega: (\mathbb Z/p\mathbb Z)^\times\rightarrow \mathbb C_p^\times$, which maps $a\in(\mathbb Z/p\mathbb Z)^\times$ to the unique root of $X^{p-1}-1$ in $\mathbb Z_p$ reducing to $a$ mod $p$, how can we consider $\omega$ as a complex character? Washington says that the Teichmuller character "may be regarded as coming from a complex character if desired, but the choice is noncanonical and depends on an embedding of $\mathbb Q(\zeta_{p-1})$ into $\mathbb Q_p$". Is the idea that we'd define the (complex) Teichmuller character to just send $a$ to a complex root of $X^{p-1}-1$. (Which root? I'd guess fixing $\zeta=\zeta_{p-1}$ primitive then setting $a\mapsto \zeta^a$?). This would then lead us to define the $p$-adic character by embedding $\mathbb Q(\zeta_{p-1})$ into $\mathbb Q_p$... Is this the right idea?

EDIT: My definition of the complex Teichmuller character doesn't work, as $\omega (ab)\neq \omega(a)\omega(b)$. But, if we send a generator of $ (\mathbb Z/p\mathbb Z)^\times$ to $\zeta_{p-1}$ then this should work.

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    $\begingroup$ Since the cardinality of $\Bbb C$ and that of $\Bbb C_p$ are both continuum, the transcendence degree over $\Bbb Q$ is continuum in both cases. From a bijection between the transcendence bases, you can pass to an isomorphism first between the associated purely transcendental subfields; then extend to algebraic closure(s) of these using the universal property of algebraic closure. You see that there’s such heavy use of Axiom of Choice that you’d never try to see an explicit definition of the final map, except to say that it was horribly discontinuous. $\endgroup$ – Lubin Feb 24 at 22:03
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    $\begingroup$ And you can send the Teichmuller character to any coherent sequence of primitive roots $\zeta_{nm}^m = \zeta_n \in \mathbb{C}$, since there is always an automorphism of $\mathbb{C}$ sending a coherent sequence of primitive roots to another one $\endgroup$ – reuns Feb 25 at 2:52

As you argue yourself at the very end: Let $K$ be any field whose characteristic does not divide $p-1$ and in which the polynomial $X^{p-1}-1$ splits completely (i.e. contains the full $p-1$-th roots of unity $\mu_{p-1}(K)$). Then there are $\phi(p-1)$ (totient "$\phi$") distinct group monomorphisms $(\Bbb Z/p\Bbb Z)^\times \hookrightarrow K^\times$, namely, for every choice of a primitive $\zeta_{p-1} \in \mu_{p-1}$, there is exactly one which maps $1$ to $\zeta_{p-1}$.

That is just a fancy restatement of the elementary algebra fact that under the given conditions, $\mu_{p-1}(K)$ is cyclic of order $\phi(p-1)$, and needs no axiom of choice to summon non-explicit isomorphisms $\Bbb C \simeq \Bbb C_p$ or non-explicit automorphisms of $\Bbb C$.

If one wants to do that for $K=\Bbb C$, of course one can evoke those things and call all these maps "Teichmüller maps", but since they are virtually indistinguishable, and the AOC-dependence of those constructions makes it impossible to get any extra information to distinguish them, i.e. to tell us which of them should correspond to the unique Teichmüller map $(\Bbb Z/p\Bbb Z)^\times \hookrightarrow \Bbb C_p^\times$, I fail to see the benefit of this over treating it with elementary field theory as above. And I think that is also Washington's point when he says that the choice is noncanonical.


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