Let $\mathbb{Q}_p$ be the field of $p$-adic numbers and let $K$ be an extension of $\mathbb{Q}$ of finite degree. The Wikipedia page "Tensor product of fields" states that $K \otimes_{\mathbb{Q}} \mathbb{Q}_p$ is a product of finite extensions of $\mathbb{Q}_p$, in $1-1$ correspondence with the completions of $K$ for extensions of the $p$-adic metric on $\mathbb{Q}$. Why is this true?
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2I believe this is answered here. Because I gave an answer there, and I am not an expert in this area I refrain from casting the first vote to close this as a duplicate. – Jyrki Lahtonen Apr 23 '17 at 20:21
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1This is Proposition 8.3 in Neukirch, you may use it as a reference. – Ege Erdil Apr 23 '17 at 20:55