Using the Seifert-van Kampen theorem to calculate the fundamental group of the torus, aided by illustrations from mspaint

One of my favorite theorems is the Seifert-van Kampen theorem. It’s a very handy result in algebraic topology which allows us to calculate the fundamental group of complicated spaces by breaking them down into simpler spaces. The version of the theorem I’ll be using here can be stated as follows: Let be a path connectedContinue reading “Using the Seifert-van Kampen theorem to calculate the fundamental group of the torus, aided by illustrations from mspaint”

A (mostly) purely algebraic proof of the infinitude of prime numbers

The following proof was provided/explained to me by @grassmanian on twitter. I thank him again for the support. Unfortunately, neither he nor I have found a way to show that each maximal ideal of contains a unique prime number without using the fact that is a PID, hence the mostly in the title. We beginContinue reading “A (mostly) purely algebraic proof of the infinitude of prime numbers”

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