I read Neukirch’s book “Algebraic Number Theory”, and its remark following to proposition VI.2.3, there is an assertion that natural map $\mathbb{A}_K \otimes_K L \to \mathbb{A}_L$ is isomorphism. How can it be proved ?
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3$\begingroup$ Can you say what you have done so far? For example, can you think of a candidate mapping from the tensor product to the adele ring of the larger field? Are you familiar with the isomorphism from $K_v \otimes_K L$ to $\prod_{w\mid v} L_w$ for each place $v$ of $K$? $\endgroup$– KConradCommented Oct 29, 2021 at 3:31
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$\begingroup$ I have an agreement to the fact that for each place $v$ there is the isomorphism $K_v \otimes_K L \to \prod_{w|v} L_w$, however the difficulty is still existing (in my thought). The map is described by the usage of restricted products. Does tensor product commute to restricted product? $\endgroup$– AliceCommented Oct 29, 2021 at 3:50
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2$\begingroup$ Do you see how $\mathbb A_K \otimes_K L$ and $\mathbb A_L$ are both $\mathbb A_K$-modules, and that both of these are free $\mathbb A_K$-modules of (finite) rank $[L:K]$? If that is too general, think first about a concrete example like $K = \mathbf Q$ and $L = \mathbf Q(i)$. The point is that to prove a ring homomorphism $A \to B$ is an isomorphism where $A$ and $B$ are not just rings but also $R$-modules for a ring $R$, think linearly and try to show the mapping is an $R$-module isomorphism. $\endgroup$– KConradCommented Oct 29, 2021 at 4:22
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This is Theorem 1 in Chapter IV-1 of Weil: Basic Number Theory. See also Corollaries 1-2 after the proof of this theorem.
answered Oct 29, 2021 at 16:44