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.



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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.