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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.

9 votes

Ways to prove the fundamental theorem of algebra

There's a linear algebra proof by Harm Derksen: https://www.jstor.org/stable/3647746. You can also find the article posted here: https://math.berkeley.edu/~ribet/110/f03/derksen.pdf
The Amplitwist's user avatar
6 votes

books well-motivated with explicit examples

Milne's lecture notes contain many good, standard examples discussed in depth. For example, in Algebraic Number Theory, in the section about Frobenius elements, Milne proves quadratic reciprocity (whi …
Anonymous's user avatar
  • 889
5 votes

Books you would like to see translated into English

The Collected Work of Carl Ludwig Siegel.
Anonymous's user avatar
  • 889
28 votes

Theorems with unexpected conclusions

The Taniyama-Shimura conjecture (now proved, by Wiles and others): all elliptic curves over $\mathbb Q$ are modular. It's magical that one can give a "formula" for the numbers of points on the curve m …
Anonymous's user avatar
  • 889
12 votes

Books you would like to see translated into English

Don Zagier's German book about quadratic forms.
Anonymous's user avatar
  • 889