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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
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Algebraic theorems with no known algebraic proofs
The fundamental theorem of algebra states that every non-constant single-variable polynomial with coefficients in $\mathbb C$ has at least one complex root; i.e., the field of complex numbers is alg …