From franzl@csusm.edu Thu May 16 17:47:20 2002
Date: Thu, 9 May 2002 15:17:47 -0700 
From: Franz Lemmermeyer <franzl@csusm.edu>
To: "'kedlaya@math.berkeley.edu'" <kedlaya@math.berkeley.edu>
Subject: class field theory notes


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Hi Kiran,

I just came across your class field theory notes and have a comment on
the
Kummer theory chapter (I've not yet gone through the whole thing).

First, I don't think that Kummer ever proved that cyclic extensions of
fields
with the appropriate roots of unity are Kummer extensions; Kummer studied
extensions Q(\zeta_p, a^1/p) (in particular, he proved Hilbert 90 for
these guys),
but he never did more than he absolutely had to in order to prove his
reciprocity
law (and FLT). When Hilbert cleaned up his work, the theory became known
as
Kummer theory, and the theorem you refer to (as well as some others, like
criteria for Kummer extensions to be abelian over the base field) was
probably
proved first by Weber, notably in his proofs of Kronecker-Weber. There
was
no Galois theory in the 1840s and 50s when Kummer did his research
(except
stuff as in the cyclotomy chapter in Gauss's Disquisitiones): Galois'
papers were
published in 1846, and only in the 1860s did Galois theory become a
subject
for university courses and books.

Next, the `general case' of Hilbert 90 is _not_ due to Emmy Noether. She
published
these results in her article on the principal genus theorem and
explicitly credited
the result to Andreas Speiser (he apparently even looked at
H^1(G,GL_n(L))).

Also, check your remark after the proof of `Kummer reformulated': if L/K
is
Z/nZ, then L(\zeta_n)/K(\zeta_n) is Z/mZ for m|n - is this what you
wanted to say?

thanks for sharing the notes,

    franz