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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?

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    @Matt, further, you can show any finite integral domain is a field without Wedderburn (see Dummit&Foote page 228).2011-12-03

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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 $\mathbb C$; ...