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