Abstract Algebra: Theory and Applications · Problem 22.4.17

The key insight is that separability is about minimal polynomials having no repeated roots, and this property is inherited by larger fields. When $K$ is separable over $F$, every element of $K$ has a minimal polynomial over $F$ with distinct roots. Since any element of $K$ also lies in $E$, its minimal polynomial over $E$ must divide its minimal polynomial over $F$—and a polynomial with distinct roots can only have divisors with distinct roots. The hint reminds you that coefficients in $F$ are automatically in $E$, so you're not losing information about the original minimal polynomial.

The key insight is that separability is about minimal polynomials having no repeated roots, and this property is inherited by larger fields. When $K$ is separable over $F$, every element of $K$ has a minimal polynomial over $F$ with distinct roots. Since any element of $K$ also lies in $E$, its minimal polynomial over $E$ must divide its minimal polynomial over $F$—and a polynomial with distinct roots can only have divisors with distinct roots. The hint reminds you that coefficients in $F$ are automatically in $E$, so you're not losing information about the original minimal polynomial.