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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 $\prod_{\mathfrak P|\mathfrak p}L_\mathfrak P$.

  1. What can we say about $\mathbb Q^c\otimes_K K_\mathfrak p$? 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.

  1. But what is an integral element of $\mathbb Q^c\otimes_K K_\mathfrak p$ over $\mathcal O_{K,\mathfrak p}$? How does such an element actually look like?

Many thanks in advance.

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