NUMBER THEORETIC BACKGROUND
5
be the map given by cup product with aE/F. Since CE, -
C%al(E/E/)
is a bijection for
F c E' c E, the property (W3) above implies, via an abstract cohomological
theorem (combine the corollary of p. 184 of [AT] with Theorem 12, p. 154, of [SI]),
that ctn(E/F) is an isomorphism for every n. Moreover, the canonical classes are
interrelated by
(1.2.4) infl (xE'/F = [E:E']aE/F and rtsaE/F = ocE/E
(for the first, use Theorem 6 on p. 188 of [AT]; the second is obvious). Thus, implicit
in the existence of Weil groups is all the cohomology of class field theory.
For example, taking n = - 1 in (1.2.3) we find
Hl(Gsd(E/F),
CE) = 0. Taking
n = 0, we find
H2(Ga\(E/F),
CE) is cyclic of order [E:F], generated by aE/F. Taking
n = - 2 we find an isomorphism
GEh/F
**
CF/NE/FCE
which, by (W^, is that given
by the reciprocity law. For E/F cyclic, this isomorphism determines aE/F,
a n
d it
follows that aE/F i s t n e "canonical" or "fundamental" class of class field theory.
The same is true for arbitrary E/F as one sees by taking a cyclic EJF of the same de-
gree as E/F, and inflating aE/F a n d ccEl/F to EEX/F, where they are equal by (1.2.4).
Conversely, if we are given classes aE/E' satisfying (1.2.4) and such that the maps
(1.2.3) are isomorphisms, then we can construct a Weil group WF as the projective
limit of group extensions WE/F made with these classes. This construction is ab-
stracted and carried out in great detail in Chapter XIV of [AT]. The existence of
such classes aE/E is proved in [AT] and [CF].
Thus, a Weil group exists for every F; to what extent is it unique?
(1.3) Unicity. A Weil group for F/F is unique up to isomorphism. More precisely:
(1.3.1)
PROPOSITION.
Let WF and WF be two Weil groups for F/F. There exists an
isomorphism 0: WF ^* WF such that the diagrams
WF Wf
wF,
(wEyh
are commutative.
For each finite Galois E/F, let 1(E) denote the set of isomorphisms/such that the
following diagram is commutative
0 CE WE/F Gol(E/F) 0
id / id
0 CE WE/F Gal(E/F) 0
Since the two group extensions WE/F&nd WE/F each have the same class, namely
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