Let then clearly so if is the minimal polynomial for over we have that . Now that means that is a basis for over . If we can show that is linearly independent then it would means because the dimension of a linearly independent set cannot the dimension of a basis. To show that is linearly independent over it is equivalent to showing that no element in this set can be expressed as a linear combination of the other elements. Take for example, , we cannot express as a linear combination where because so it is impossible to get intermediately exponents between and . Thus, and this means .

This is Mine 87th Post!!!