1) Suppose we have some field $F$ then it is known that the smallest subfield of $F$ called $F_0$ say is given by the intersection of all subfields of $F$.
Is the reason for this because every family of subfields is a subfield so the intersection of all subfields is a subfield so any subfield contains this intersection hence that is the smallest subfield?
2) What does $F(u_1,...,u_n)$ mean? I have seen that it is the smallest field containing $F$ and the elements $u_1,...,u_n$ so would that just be the intersection of all the subfields of $F$ along with the elements $u_1,...,u_n$?
I have also seen it defined as the field of fractions i.e the field of $f/g$ where $f,g$ have coefficients $F(u_1,...,u_n)$ are these exactly the same concepts I'm having trouble seeing that there are the same.
3) What is $F[x_1,x_2,...,x_n]$? I that $F[x]$ is the set of all polynomials with coefficients in $F$ but I don't know what switching the $[x]$ to $[x_1,...,x_n]$ actually means?
I have tried to explain what I know and would just appreciate if someone could help me with the notation which is halting my progression further into algebra.