Let $L/K$ be an extension of number fields, $\frak P|\frak p$ primes of $\mathcal O_L$ and $\mathcal O_K$, and let $K_\frak p$ be the completion of $K$ wrt $|\cdot|_\frak p$ and $L_\frak P$ the completion of $L$ wrk to $|\cdot|_\frak P$.
We can write $L=K(\alpha)$. Write the minimum polnomial $f(X)$ of $\alpha$ over $K$ as $f(X)=f_1(X)\cdot\cdot\cdot f_r(X)$ where $f_i(X)\in {K_\frak p}[X]$ are distinct irreducibles.
We then have an isomorphism of $K_\frak p$-algebras $$L\otimes_K K_{\frak p}\cong \bigoplus_{i=1}^r K_{\frak p}[X]/(f_i(X))$$
by the Chinese Remainder Theorem.
My question is:
What can we say about the topological aspect of this isomorphism?
For example, is it possible to define $|\cdot|_\frak p$-norms on both sides so that the isomorphism is continuous?
Many thanks for your help.
The spaces which the RHS comprises are complete local rings which extend the base ring, so any absolute value you choose will be unique and determined by the usual formula involving the norm, i.e.
In particular, it's well-known that there is only one choice of topology you can put on the RHS to make it compatible with the topolgical vector space strucuture, the product topology, which is equivalent to the sup norm when you treat this as a product of $K_{\mathfrak{p}}$ vector spaces. We easily see this is just the sup norm on the vector of components with each component having the unique extension absolute value.
But then the map need only be tested in this topology. In fact, if you look at the proofs in some texts (eg Weil's Basic Number Theory p.48), you'll see that this is made specifically to be a topological algebra isomorphism in a unique way. But we can be explicit without having to worry about the uniqueness: this is a linear map of vector spaces, so continuity is immediate from linearity and finite dimensionality.