Can anyone cite an example of a mathematics paper that has been retracted?


It is said that on the order of 100,000 new theorems enter the mathematics literature every year. For a number of reasons including hyper-specialization and demands on referee resources it is, in my view, unlikely that all their proofs are correct. Yet it seems no explicit effort is made to clean the literature. False theorms float downstream along with the true ones, available for citation and use in constructing yet further theorems.


Thanks for any insight.


Cheers, Scott



Retracted Mathematics Papers
A general existence theorem is proved :


1933 : W. Grunwald, *Ein allgemeines Existenztheorem für algebraische Zahlkörper*, J. reine angew. Math. 169 (1933), 103–107.


and reproved :


1942: G. Whaples, *Non-analytic class field theory and Grünwald's theorem*. 
Duke Math. J. 9, (1942). 455–473. 


A counter-example is found :


1948 : S. Wang, *A counter-example to Grunwald's theorem*, Ann. Math. 49 (1948), 1008–1009. 


and the theorem is corrected :


1950 : S. Wang, *On Grunwald's theorem*, Ann. Math. 51 (1950), 471–484.


twice in the same year :


---- : H. Hasse, *Zum Existenzsatz von Grunwald in der Klassenkörpertheorie*, J. reine angew. Math. 188 (1950), 40–64.


A quarter of a century later, a simpler proof is given :


1974: J. Neukirch, *Eine Bemerkung zum Existenzsatz von Grunwald-Hasse-Wang*, J. Reine Angew. Math. 268/269 (1974), 315–317. 


but more than half a century later, corrections to the corrections are required :


2007 : W-D. Geyer & C. Jensen, *Embeddability of quadratic extensions in cyclic extensions*.
Forum Math. 19 (2007), no. 4, 707–725. 


2011 : P. Morton, *A correction to Hasse's version of the Grunwald-Hasse-Wang theorem*. 
J. Reine Angew. Math. 659 (2011), 169–174. 


**Addendum** (2013/05/18)


I'm afraid the above list of errors and corrections might look a bit negative, so let me add a positive note (which will also save you 30,00 € or $42.00 by not having to read it [here](http://www.degruyter.com/view/j/form.2007.19.issue-4/forum.2007.027/forum.2007.027.xml)) :


In 1933, van der Waerden asked in the Jahresbericht : Which quadratic fields can be embedded in cyclic quartic fields ? Solutions were provided by four people, among them Hasse, who generalised the problem to : Under which conditions can a degree-$l$ ($l$ prime) cyclic extension $K\_1$ of a number field $K$ be embedded into a degree-$l^n$ cyclic extension $K\_n$ of $K$ ?


A. Scholz sent in a "solution" to this problem in 1935 which essentially claimed that the obstructions are purely local in nature. But Hans Richter, a doctoral student of van der Waerden, knew already that there is an exception when $l=2$, so a Scholtz-Richter correction to Scholz's paper was required. In a sense, Richter anticipated not only Wang's counterexample to Grunwald's theorem but also its solution, without mentioning it explicitly as such.


