Number of Q-embeddings into C Let $\alpha $ be a root of the polynomial ${x^5} + 6{x^3} + 8x + 10$. How many $\mathbb{Q}$-embeddings of $\mathbb{Q}\left[ {\alpha ,\sqrt 7 } \right]$ (the least field extension of $\mathbb{Q}$ which contains elements $\alpha $ and $\sqrt 7 $) into $\mathbb{C}$ does there exist?
 A: Since $p\left( x \right) = {x^5} + 6{x^3} + 8x + 10$ is irreducible over $\mathbb{Q}$ by Eisenstein criterion, $p$ is minimal polynomial for $\alpha $ over $\mathbb{Q}$, from which it follows that $\left[ {\mathbb{Q}\left[ \alpha  \right]:\mathbb{Q}} \right] = 5$. Furthermore, $\sqrt 7  \notin \mathbb{Q}\left[ \alpha  \right]$, otherwise we would have $\underbrace {\left[ {\mathbb{Q}\left[ \alpha  \right]:\mathbb{Q}} \right]}_{ = 5} = \left[ {\mathbb{Q}\left[ \alpha  \right]:\mathbb{Q}\left[ {\sqrt 7 } \right]} \right]\underbrace {\left[ {\mathbb{Q}\left[ {\sqrt 7 } \right]:\mathbb{Q}} \right]}_{ = 2}$ which is impossible.  So, $\left[ {\mathbb{Q}\left[ {\alpha ,\sqrt 7 } \right]:\mathbb{Q}\left[ \alpha  \right]} \right] \geqslant 2$. Also, ${x^2} - 7 \in \left( {\mathbb{Q}\left[ \alpha  \right]} \right)\left[ x \right]$ (polynomials with coefficients in ${\mathbb{Q}\left[ \alpha  \right]}$) which gives us $\left[ {\mathbb{Q}\left[ {\alpha ,\sqrt 7 } \right]:\mathbb{Q}\left[ \alpha  \right]} \right] \leqslant 2 \Rightarrow \left[ {\mathbb{Q}\left[ {\alpha ,\sqrt 7 } \right]:\mathbb{Q}\left[ \alpha  \right]} \right] = 2$.
We conclude that $\left[ {\mathbb{Q}\left[ {\alpha ,\sqrt 7 } \right]:\mathbb{Q}} \right] = \left[ {\mathbb{Q}\left[ {\alpha ,\sqrt 7 } \right]:\mathbb{Q}\left[ \alpha  \right]} \right]\left[ {\mathbb{Q}\left[ \alpha  \right]:\mathbb{Q}} \right] = 2 \cdot 5 = 10$.
So, there are 10 different $\mathbb{Q}$-embeddings of ${\mathbb{Q}\left[ {\alpha ,\sqrt 7 } \right]}$ into $\mathbb{C}$.
