Let \(E\) be an intermediate field extension in \(K \leq E \leq F\).
Show that if \(u\in F\) is separable over \(K\), then \(u\) is separable over \(E\).
Show that if \(F\) is separable over \(K\), then \(F\) is separable over \(E\) and \(E\) is separable over \(K\).
Solution
1. If \(u\in F\) is separable over \(K\), then \(u\) is separable over \(E\).
Let \[ m_K(x)\in K[x] \] be the minimal polynomial of \(u\) over \(K\), and let \[ m_E(x)\in E[x] \] be the minimal polynomial of \(u\) over \(E\). Since \(K[x]\subseteq E[x]\) and \(m_K(u)=0\), minimality of \(m_E\) implies \[ m_E\mid m_K\qquad\text{in }E[x]. \]
Because \(u\) is separable over \(K\), the polynomial \(m_K\) has no repeated root in a splitting field. Every divisor of a polynomial with distinct roots likewise has distinct roots. Hence \(m_E\) is separable, so \(u\) is separable over \(E\).
2. If \(F/K\) is separable, then \(F/E\) is separable.
Let \(u\in F\). Since \(F/K\) is separable, \(u\) is separable over \(K\). By 1, \(u\) is separable over \(E\). Since this holds for every \(u\in F\), the extension \(F/E\) is separable.
3. If \(F/K\) is separable, then \(E/K\) is separable.
Every element of \(E\) is also an element of \(F\). Since every element of \(F\) is separable over \(K\), every element of \(E\) is separable over \(K\). Thus \(E/K\) is separable.