From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: square in every Z_m implies square?
Date: 15 May 1998 15:19:25 GMT
In article <6jdpm4$o6j$1@vixen.cso.uiuc.edu>,
adam louis stephanides wrote:
>Bill Dubuque writes:
>
>>Simpler: it's an instance of the Hasse local-global principle.
>
>Don't you also need solvability in R to apply that?
Very perceptive of you -- the local-global principle, when applied to a
number field, require local solvability in _all_ completions of a field.
But for the rational number field, it's sufficient to work only in all
p-adic completions; I quote exercise 3.6 from Cassels' "Lectures on
Elliptic Curves":
Do you observe anything about the parity of the number N of
primes (including \infty) for which there is insolubility? If not,
construct similar exercises [to numerical problems in exercise 5]
and solve them until the penny drops.
Likewise, Borevich&Shafarevich prove the Hasse-Minkowski Theorem
(section 1.7.2) and remark that one never needs to verify solvability
at p=2 in order to deduce global solvability.
dave