Let $D$ be a finite central division algebra over $Z$ (the center of $D$), and $F$ be a maximum subfield of $D$ (that is to say, there does not exist a larger subfield $G$ of $D$ such that $G$ properly includes $F$). Then $F$ can be viewed as a vector space over $Z$.
The question is:
Is $\dim_ZF$ a certain number?
I know that $F$ may be not unique. But is the dimension of $F$ over $Z$ an invariant?
It seemed that Emil Artin had studied this problem, but I do not understand the details of the problem.
Yet i still do not understand it,but I find the answer is $\textbf{YES}$.See below:
$\textbf{Comments:}$
1.I am pretty sorry to make you confused about $\textbf{Z}$ and I clarified it just now.It means the center of $\textbf{D}$,and none of integer's business.
2.Nathan Jacobson is a master of ring theory,especially non-commutative ring.We can read his books to search for the answer,it is referred in his $\textbf{BASIC ALGEBRA 2}$,and it starts from $\textbf{PAGE 215,CHAPTER 4.6,FINITE DIMENSIONAL CENTRAL ALGEBRAS}$.May be $\textbf{DENSITY THEOREM}$ is also required and it is talked about in $\textbf{CHAPTER 4.3}$.
3.I do not know how to upload a pdf file to this website,it will be kind of you to help.If you want,I can share E-book of $\textbf{BASIC ALGEBRA 1 && 2}$ to you.
4.I do not know the essential parts.It is my first time to learn it,and it seems tough to me.
see http://www.math.northwestern.edu/~len/d70/chap17.pdf.It suffices to answer this problem,and it is really a nice textbook to learn $\textbf{Simple rings}$.Thank you for your views.
I suppose $Z$ is just the name of the center of $D$, which is a field. The dimension $\dim_Z(D) = n^2$ is always a square, and $\dim_Z(F) = n$ holds for every maximal subfield $F$ of $D$. Details can be found e.g. in [Farb, Dennis - Noncommutative Algebra].