For a number field $K/\mathbb{Q}$, we say that a finite place of $Q$ is ramified if there exists a valuation $v_{p_i}$ in $K$ lying over $v_p$ such that it is ramified in the sense of the associated discrete valuation rings. This is the ramification index, denoted $e_{p_i/p}$.

We also have the residue field extension, of degree $f_{p_i/p}$.

With these, we can define $n_{p_i/p}$ as $f_{p_i/p}e_{p_i/p}$, and the $n_{p_i/p}$ behave well, they sum to the degree of the extension and so on.

Now this is all lovely over a finite place, but for infinite places, in the treatment I saw, only the integers $n_{\infty_i/\infty}$ were defined, as $1$ if the associated extension is real to real, and $2$ if real to complex. These infinite $n_{\infty_i/\infty}$ behave in the same way as the finite $n_{p_i/p}$.

So then the question is, why do we view $n_{\infty_i/\infty}=2$ as ramified, eg, from the perspective of CFT. Is there an intuitive explanation of why these ought to be ramified and not just the analogue of a purely $f_{p_i/p}$ extension of finite places?


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