- #1
mathmari
Gold Member
MHB
- 5,049
- 7
Hey!
Let $F\subseteq E\subseteq K$ be consecutive field extensions and $f\in F[x]$ be non-constant.
I want to show that if $K$ is the splitting field of $f$ over $F$, then $K$ is the splitting field of $f$ also over $E$.
Since $K$ is the splitting field of $f$ over $F$, we have that $f(x) = c\prod (x-a_i)^{m_i}$, where $m_i$ are non-negative integers and $(x-a_i)\in K[x]$.
Since $F\subseteq E$ we have that $f\in E[x]$. Therefore, $K$ must be also the splitting field of $f$ also over $E$.
Is this correct? (Wondering)
Let $F\subseteq E\subseteq K$ be consecutive field extensions and $f\in F[x]$ be non-constant.
I want to show that if $K$ is the splitting field of $f$ over $F$, then $K$ is the splitting field of $f$ also over $E$.
Since $K$ is the splitting field of $f$ over $F$, we have that $f(x) = c\prod (x-a_i)^{m_i}$, where $m_i$ are non-negative integers and $(x-a_i)\in K[x]$.
Since $F\subseteq E$ we have that $f\in E[x]$. Therefore, $K$ must be also the splitting field of $f$ also over $E$.
Is this correct? (Wondering)