Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field containing $K$ then $L\otimes_K K_\mathfrak p$ is a product of fields $\prod_{\mathfrak P|\mathfrak p}L_\mathfrak P$

What can we say about $V=\mathbb Q^c\otimes_K K_\mathfrak p$? Is it also a product of fields?

Also in his book 'Galois Modules' Frohlich defines the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$ as the integral closure of $\mathcal O_{K,\mathfrak p}$ inside the tensor product.

Presumably $\mathcal O_{K,\mathfrak p}$ sits inside $\mathbb Q^c\otimes_{\mathcal O_K} \mathcal O_{K,\mathfrak p}=V$ but how would an element of $\mathbb Q^c\otimes_K K_\mathfrak p$ integral over $\mathcal O_{K,\mathfrak p}$ actually look like?

Many thanks for your help.


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