Dimension of vector space generated by finite field over its subfield

Question

Question: let $F$ be field of order $7^6$ and let $H$ be it's subfield of $F$ containing $49$ elements, then dimension of vector space form by $F$ over $H$ is?

I just know, every field form a vector space over its subfield. But from this we can't determine dimension. I had seen some familiar examples like, $dim(\mathbb{R}^3(\mathbb{R}))= 3$ etc. But here, it can't works, is there is any formula or any method? Please help me..

Answer 1

Hint:

For any prime $\;p\;$ and natural numbers $\;n,\,m\;$ such that $\;m\,\mid \,n\;$ , we have that

$$\dim\left(\Bbb F_{p^n}/\Bbb F_{p^m}\right)=\frac nm$$

Answer 2

Let $\beta=\{x_1,x_2,x_3,...x_n\}$ be the basis we need to find $n$. Possible linear combinations $=7^2\times7^2...\times7^2=7^{2n}=7^6\implies n=3$