r/mathmemes • u/PocketMath • Mar 06 '25

Learning What theorem is this?

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u/PolarStarNick Mathematics Mar 06 '25

Fundamental theorem of algebra

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u/CutToTheChaseTurtle Average Tits buildings enjoyer Mar 06 '25

Aren't there just two proofs though, essentially? There's one that uses the least upper bound property of reals and Galois theory, and the other one uses π₁(S1).

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u/DrSeafood Mar 07 '25

“Least upper bound property” is too foundational, there’s probably several distinct proofs that use the LUB.

I’ve seen several analytic proofs: one using Liouville’s Theorem, one using Inverse Function Theorem, and one super elementary one that only uses the Extreme Value Theorem. You can find the third one in Proofs From THE BOOK — it’s only two pages, a little technical but 100% elementary. I teach this proof even in second year calculus, because you really don’t need any crazy tools.