Let $E$ be a finite extension of the field $\mathbb { Q }$. We say that a homomorphism of fields $\phi : E \longrightarrow \mathbb { C }$ is real if $\phi ( E ) \subset \mathbb { R }$. Prove or disprove each of the following assertions: (A) For each prime number $p$, the field $\mathbb { Q } \left( p ^ { 1 / 12 } \right)$ has exactly one real embedding in $\mathbb { C }$. ($p ^ { 1 / 12 }$ is the unique real number $r > 0$ such that $r ^ { 12 } = p$.) (B) If $[ E : \mathbb { Q } ] = 11$, there exists a real embedding of $E$. (C) If $E$ is a Galois extension of $\mathbb { Q }$ and $[ E : \mathbb { Q } ] = 11$, then every embedding $E \longrightarrow \mathbb { C }$ is a real embedding.
Let $E$ be a finite extension of the field $\mathbb { Q }$. We say that a homomorphism of fields $\phi : E \longrightarrow \mathbb { C }$ is real if $\phi ( E ) \subset \mathbb { R }$. Prove or disprove each of the following assertions:\\
(A) For each prime number $p$, the field $\mathbb { Q } \left( p ^ { 1 / 12 } \right)$ has exactly one real embedding in $\mathbb { C }$. ($p ^ { 1 / 12 }$ is the unique real number $r > 0$ such that $r ^ { 12 } = p$.)\\
(B) If $[ E : \mathbb { Q } ] = 11$, there exists a real embedding of $E$.\\
(C) If $E$ is a Galois extension of $\mathbb { Q }$ and $[ E : \mathbb { Q } ] = 11$, then every embedding $E \longrightarrow \mathbb { C }$ is a real embedding.