Let $L$ and $M$ be two finite extensions of $\mathbb{Q}$ and let $LM$ denote their compositum. Suppose that $p$ is a rational prime that splits completely in $L$ and $M$. How can I show that $p$ splits completely in $LM$?
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are these Galois extensions or just general ones? – Adam Hughes Apr 28 '16 at 18:42
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General ones, not necessarily Galois. – joy Apr 28 '16 at 18:45
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As $p$ splits completely, we have $\mathbb Q_p\otimes_{\mathbb Q}L\cong\mathbb Q_p\oplus\dots\oplus\mathbb Q_p$ (isomorphism of $\mathbb Q_p$-algebras, $[L:\mathbb Q]$ copies of $\mathbb Q_p$) and likewise $\mathbb Q_p\otimes_{\mathbb Q}M\cong\mathbb Q_p\oplus\dots\oplus\mathbb Q_p$. Now $LM$ is a subfield of the $\mathbb Q$-algebra $L\otimes_{\mathbb Q}M$, so $\mathbb Q_p\otimes_{\mathbb Q}LM$ is a subalgebra of $(\mathbb Q_p\otimes_{\mathbb Q}L)\otimes_{\mathbb Q_p}(\mathbb Q_p\otimes_{\mathbb Q}M)\cong \mathbb Q_p\oplus\dots\oplus\mathbb Q_p$, i.e. it is again isomorphic to $\mathbb Q_p\oplus\dots\oplus\mathbb Q_p$, i.e. $p$ completely splits in $LM$.
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