Let be a field and let be an extension of . Let be algebraic over with minimal polynomials and , respectively. Show that .
Recall that has a kind of universal property with respect to fields containing an element whose minimal polynomial over is . Since contains an element which is algebraic over with minimal polynomial , we have an injective ring homomorphism fixing and . Similarly, we have an injective map fixing , , and . Since every element of is a linear combination of , this map is the identity. So .