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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.

Finally,

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.

Thanks!

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Question 1. If $\{F_\alpha\}$ is the collection of all subfields of a field $F$, it is clear that as $\bigcap_\beta F_\beta\subset F_\alpha$ (as a subset) for all $\alpha$. Now if $x\in \bigcap_\beta F_\beta$, then $x\in F_\alpha$ for all $\alpha$. Then $x^{-1}\in F_\alpha$ for all $\alpha$. Meaning $x^{-1}\in \bigcap_\beta F_\beta$. Similarly one check that if $x,y\in \bigcap_\beta F_\beta$, then so are $x+y$ and $xy$. This shows $\bigcap_\beta F_\beta$ is a subfield of $F$ and it is the smallest since it is contained in every subfield.

Question 2. This is called transcendental extension of a field. See for example here to find the answer to all of your questions.

You have, by the way, a confusion: $F(u_1, \cdots, u_n)$ is the smallest field containing $F$ (not contained in $F$, it is not a subfield of $F$) and $u_1, \cdots, u_n$. In fact $F(u_1, \cdots, u_n)$ is much much bigger than $F$. As an example think about $\mathbb{C}$ and the field $\mathbb{C}(x)$ which is $$ \mathbb{C}(x)=\left\{\left.\frac{p(x)}{q(x)}\right| p(x), q(x)\text{ are polynomials in }\mathbb{C}, q(x)\neq 0\right\} $$ This the the field of rational functions on $\mathbb{C}$. Obviously $\mathbb{C}\subset \mathbb{C}(x)$. If you think about it a little this field is also the smallest field containing $\mathbb{C}$ and $x$. Try to figure out why in this simple example, then read that note I told you for a generalization. Hint: try to show that the smallest field containing $x$ and $\mathbb{C}$ contains all polynomials $a_1+a_2x+\cdots+a_nx^n$. Then if it does so, since this is a field, any non-zero element has an inverse, so it should also contain $1/(a_1+a_2x+\cdots+a_nx^n)$.

Question 3. Yes, $F[x_1, \cdots, x_n]$ is the polynomial ring in indeterminant $x_1, \cdots, x_n$ with coefficients in $F$. The ring $F[x]$ is the polynomial ring in only one indeterminant. For example $$ 1+ x^4 + 87 x^{87}\in \mathbb{Q}[x], \quad 2016+x^2+y^2+ 65 x^7y^{43}\in \mathbb{Q}[x,y] $$ The only difference between $F[x]$ and $F[x_1, \cdots, x_n]$ is the number of variables involved.

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