Question

Let F, K, and L be fields where F⊂K⊂L. If L is a finite extension of F and [ L:F]=[L:K], prove that F = K. Show all your work, including the necessary steps and reasoning to prove this statement.

Answer 1

To prove that F = K, we need to show that every element in F is also in K, and every element in K is also in F.

Step-by-step explanation:

To prove that F = K, we need to show that every element in F is also in K, and every element in K is also in F.

1. Since L is a finite extension of F, we know that every element in L can be expressed as a finite sum of products of elements in F.
2. Since [L:F] = [L:K], this means that every element in K can be expressed as a finite sum of products of elements in F.
3. Therefore, every element in K is also in F, and by extension, every element in F is also in K. Hence, F = K.