The first proofs of class field theory used substantial analytic methods.

Two proof of concept versions were built, different in size and detail but using the same layout.

Its proof does not use the random oracle model.

The proof for the general case uses a similar method.

The proof uses a particular fact about computable real numbers.

The proof uses diagram chasing arguments similar to that above.

To give a flavor of how the proofs used to derive the properties in the previous section work, let us show the following fact.

A probabilistic proof uses the weak law of large numbers.

Although the proof uses probability, the final conclusion is determined for certain, without any possible error.

The proof uses the nonsolvability of the symmetric group S5.