Applications Wedderburn's Little Theorem I've been asked to give a short presentation of Wedderburn's Theorem that every finite domain is a field.  
However, the proof itself is quite short so I thought to add some applications (since this theorem doesn't lend itself to giving many examples beyond "here is a finite domain, its a field!"). What are some interesting applications of this theorem?
 A: Warning: In this answer, I'll comment about things a lot of MSE users know much better than I. I hope they'll correct and/or complete this answer. 
In André Weil's book Basic Number Theory.
Wedderburn's Little Theorem is used in an essential way to compute the Brauer group of a local field. 
The use of WLT is at the same time conspicuous and hidden. 
It is conspicuous because WLT is Theorem 1 of Chapter 1, and is stated on top of page 1. 
It is hidden because WLT is never (as far as I can see) referred to explicitly in the sequel. 
But if one looks closely, one sees that it is implicitly used a lot of times. 
The first time is in Corollary 1 to Theorem 2 page 2. 
[In this post, division rings will be called "fields", that is "not necessarily commutative fields", to stick to Weil's terminology.] 
WLT is tacitly used to conclude that the residue field of a (not necessarily commutative) non-archimedian local field is a finite commutative field. 
This enables Weil to describe in a very precise way the structure of such a (not necessarily commutative) non-archimedian local field, viewed as a division algebra over its center: see Proposition 5 page 20. 
This Proposition is the culminating point of Chapter 1, and then it is used in a crucial way in: 


*

*the comment following Definition 6 page 184,

*the proof of Theorem 1 page 222, 

*the proof of Corollary 1 page 223. 
A particularly important excerpt is the paragraph just before Theorem 2 page 224. Here is a part of this paragraph: 

As we identify the Brauer group $B(K)$ with the group $H(K)$ considered in theorem 1 and its corollaries, we may consider the mapping $\eta$ defined in corollary 2 of th. 1 as an isomorphism of $B(K)$ onto the group of roots of $1$ in $\mathbb C$; ...

