Short mathematical proofs for teaching I am looking for short proofs in order to illustrate undergraduate notions. Most of the time students struggle with technical exercises without having the time, before going on with the following semester, to realize some great applications or insights about the objects introduced. 
I would like to have some such topics to present in detail, in between 10 and 30 minutes on blackboard. To show some examples in my mind:


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*Poisson formula used to prove Minkowski theorem

*Dimension of spaces of modular forms

*Equiperimetric inequalities

*Prime Number Theorem (or weakened versions) by analytic methods


I would like to develop these themes in many other directions: group actions, representation theory, complex analysis, algebraic number theory, local inversion, PDEs, etc.

For a class on abstract algebra, there is the proof that the complex numbers are an algebraically closed field, following Artin. This is very short and can be presented in $10$ to $30$ minutes on the blackboard. One obtains it as a "great application" of Galois theory. For details, see also here:
Is there a purely algebraic proof of the Fundamental Theorem of Algebra?
