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For example, the Weber's proof of Kronecker–Weber theorem. I would like to know such proofs. It seems to be important for me to remember that a widely accepted proof might be wrong.

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    $\begingroup$ Can we ask that answers in mathoverflow.net/questions/879 not be repeated here? (Unfortunately, that probably won't work.) $\endgroup$ – GEdgar Aug 11 '13 at 1:24
  • $\begingroup$ I. R. Shafarevich, Construction of fields of algebraic numbers with given solvable Galois group (1954) math.stackexchange.com/questions/25242/… $\endgroup$ – Makoto Kato Aug 11 '13 at 3:47
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    $\begingroup$ Pontryagin proved that $\pi_{n+2}(\mathbb{S}^n) = 0$ in 1938. However, he and G.W. Whiteheand corrected it in 1950 independently. $\endgroup$ – Makoto Kato Aug 11 '13 at 3:48
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There are many variations on this theme in MathOverflow.

Widely accepted mathematical results that were later shown wrong

Failures that lead eventually to new mathematics

Most interesting mathematics mistake

How to refer to a theorem that you have shown to be wrong

What are some correct results discovered with incorrect or no proofs

Examples where physical heuristics led to incorrect answers

Oldest bug in computer algebra

Retracted mathematics papers

What mistakes did the italian algebraic geometers actually make

Italian school of algebraic geometry and rigorous proofs

Can a mathematical definition be wrong

Mathematicians whose works were criticized by contemporaries but became widely accepted later

What are examples of theorem which were once valid then became invalid as standard definitions shifted

Have we ever lost any mathematics

Examples of conjectures that were widely believed to be true but later proved false

Statements which were given as axioms which later turned out to be false

and for good measure

Smith-Minkowski-Siegel mass formula

Grunwald-Wang theorem

Polygons Flip Finitely: Flaws and a Fix

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