[FOM] Boiling down proofs (ignoring "undergraduates")
joeshipman@aol.com
joeshipman at aol.com
Fri Aug 24 13:42:13 EDT 2007
I am replying late to this "closed thread" because my membership in the
FOM list was temporariliy disabled due to oversensitivity to "bounced"
messages from AOL.
Aitken's point about the diversity of undergaduate curricula is
well-taken, so to clarify, I am more interested in the general
streamlining and simplification of mathematical proofs of particular
results, and "teachable to undergraduates" is just a shorthand for "a
proof that is not very hard and does not require a great deal of
prerequisite material".
Chow's examples of "theory of special functions" and Lie Theory from
the late 19th century are pretty good; they differ from my example of
Dirichlet's theorem in that those results have been significantly
streamlined in 100+ years, while Dirichlet's theorem remains almost as
hard as ever.
I have seen a short elementary proof of the Kronecker-Weber theorem
(another Chow example) in a Galois Theory textbook by Lisl Gaal.
However, I now understand that a later edition of that book omits the
proof, so maybe it was erroneous, in which case that theorem (any
number field with finite Abelian Galois group over Q is contained in a
field generated by a root of unity) is another good example.
The 19th-algebraic geometry results Chow refers to were not rigorously
proved until the 20th century so I don't count them. My subject is
"boiling down" valid proofs.
The best example I know of a REALLY long proof which has not been
simplified in any significant way is the Feit-Thompson Odd Order
Theorem from 1960. Can anyone suggest an earlier example?
-- JS
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