All maximal subfields of a Division Algebra are isomorphic. How can I show that two maximal subfields of a division-algebra have the same dimension over k.
I found a simple proof here on page 4:http://www.math.northwestern.edu/~len/d70/chap17.pdf. Is there a simpler way to show this with even less theory needed?
And is there a very simple way to prove it only for finite division-algebras. 
I would like to prove Wedderburns Theorem with this Lemma and the Skolem-Noether Theorem, that I already proved .  
 A: The original version of the question asked to prove that maximal subfields are isomorphic. This is wrong, see below.
Many counterexamples can be found inside the division algebra of quaternions with rational coefficients. 
Let
$$
D=\{a+bi+cj+dk\mid a,b,c,d\in\Bbb{Q}\}
$$
with the usual relations $i^2=j^2=k^2=-1=ijk$. Because $\dim_{\Bbb{Q}}D=4$ we know that all the maximal subfields are quadratic extensions of $\Bbb{Q}$.
Conversely, any field $L, \Bbb{Q}\subset L\subset D$, such that $[L:\Bbb{Q}]=2$ is maximal.
But then we have several non-isomorphic maximal subfields.


*

*Because $i^2=-1$ we see that $D$ has a copy of $\Bbb{Q}(\sqrt{-1})$
as a subfield. 

*Because $u=i+j+k$ satisfies $u^2=-3$ we also get a copy $\Bbb{Q}(u)$ of $\Bbb{Q}(\sqrt{-3})$.

*In fact, if $m=n_1^2+n_2^2+n_3^2$ we have $(n_1i+n_2j+n_3k)^2=-m$, and consequently a maximal subfield isomorphic to $\Bbb{Q}(\sqrt{-m})$. By the celebrated three squares theorem any squarefree number $m\not\equiv7\pmod8$ can be written as a sum of three squares.

*But $\Bbb{Q}(\sqrt{-n_1})$ and $\Bbb{Q}(\sqrt{-n_2})$ are non-isomorphic unless $n_1/n_2$ is a square of a rational number.



There is no shortage of non-isomorphic maximal subfields of $D$.


A "standard" proof for the fact that all the maximal subfields have the samed dimensiongoes by applying the double centralizer theorem. More precisely, the extension degree of any maximal subfield equals the square root of the dimension of the division algebra over its center. After all, any subfield is contained in its centralizer. And if the containment is proper then the field can be extended by adjoining any element from its centralizer.
