Intersection of two fields

Question

I am doing a qus and stuck at one step.... Qus- let F1 & F2 are two subfield of a finitefield F consisting of 2^9 and 2^6 elemnt...then total no. Of elemnt in F1 intersection F2.

According to me F1∩F2 is also field contained in F1 and F2 so choices of F1∩F2 is 2¹ and 2² but its answer only 2² why?

Answer 1

The nonzero elements of $F_1$ are the roots of $x^{2^9-1}=1$.

The nonzero elements of $F_2$ are the roots of $x^{2^6-1}=1$.

Therefore, the nonzero elements of $F_1 \cap F_2 $ are the roots of $x^{n}=1$, where $n=\gcd(2^9-1, 2^6-1)=7=2^3-1$.

Answer 2

A field F of order p^{n} contains another field of order p^{d} if and only if d divides n.