Question

Let F,K and L be field such that $\mathit{F}\mathbf{⊑}\mathit{K}\mathbf{⊑}\mathit{L}$. If $\left[L:F\right]$is finite then prove that $\mathbf{[}\mathbf{L}\mathbf{:}\mathbf{K}\mathbf{]}$and $\mathbf{[}\mathbf{K}\mathbf{:}\mathbf{F}\mathbf{]}$are also finite and both are $\le \mathbf{[}\mathbf{L}\mathbf{:}\mathbf{F}\mathbf{]}$.

Short answer

$\left[L:K\right]$and $\left[K:F\right]$ are finite and$\le \left[L:F\right]$.

Step-by-step solution

Definition of simple extension.

A simple extension is a field which is generated by the adjunction of single element. Every finite is a simple extension of prime field of the same characteristic.

Step 2:Showing that[L:K]and [K:F]are finite.

Let F,K and L be field such that $F⊑K⊑L$ and $\left[L:F\right]$is finite.

Since $F⊑K⊑L$ and $\left[L:F\right]$are finite. So,

$\left[L:F\right]<\infty$

As every linearly independent set of vector of L over K is also linearly independent over Fbecause $F⊑K$. Thus,

$\left[L:K\right]\le \left[L:F\right]$

And

$\left[K:F\right]\le \left[L:F\right]<\infty$

Therefore $\left[L:K\right]$ and $\left[K:F\right]$ are also finite and both are$\le \left[L:K\right]$ because K is a subspace of a finite dimensional F-vector space L .