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Pure Quintic Fields 1
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§ 3. Conductor and discriminants.
Let R = q_{1} … q_{s} be the squarefree product
of all prime divisors of the radicand D
of the pure quintic field L = Q(D^{1/5}).
Independently of the exponents e_{1},…,e_{s},
the conductor f of the cyclic quintic relative extension Nk
is given by
f^{4} = 5^{2} R^{4}
if not D ≡ ±1,±7 (mod 25) (field of the first kind),
f^{4} = R^{4}
if D ≡ ±1,±7 (mod 25) (field of the second kind)
[Thm. 1, p. 103, Ma1].
Examples 3.1.
The smallest radicands D of pure quintic fields L = Q(D^{1/5})
with given kind and increasing number of prime divisors are
D = 2, 6 = 2*3, 42 = 2*3*7, for fields of the first kind with (D,5) = 1,
D = 5, 10 = 2*5, 30 = 2*3*5 , for fields of the first kind with 5D,
D = 7, 18 = 2*3^{2}, 126 = 2*7*3^{2}, for fields of the second kind.
It is well known that the cyclotomic discriminant takes the value
d(k) = +5^{3} = 125,
and Hilbert's Theorem 39 on discriminants of composite fields shows the following
Theorem 3.1.
The metacyclic discriminant is given by
d(N) = d(k)^{5} f^{16} =
= 5^{23} R^{16} for a field of the first kind,
= 5^{15} R^{16} for a field of the second kind,
the intermediate discriminant is given by
d(M) = 5 d(k)^{2} f^{8} =
= 5^{11} R^{8} for a field of the first kind,
= 5^{7} R^{8} for a field of the second kind,
and the pure quintic discriminant is given by
d(L) = d(k) f^{4} =
= 5^{5} R^{4} for a field of the first kind,
= 5^{3} R^{4} for a field of the second kind.
§ 4. Herbrand quotient of units of Nk.
In section 2, we have seen that the unit index
(U_{N}:U_{0}) = 5^{e}
admits a coarse classification of pure quintic fields
according to the seven possible values of the exponent
0 ≤ e ≤ 6.
However, the Galois cohomology of the unit group U_{N}
with respect to the cyclic quintic Kummer extension Nk
provides additional structural information.
The Herbrand quotient of U_{N} under the action of Gal(Nk) = < σ >
is defined by
#H^{0}(Gal(Nk),U_{N}) /
#H^{1}(Gal(Nk),U_{N}) =
= #[ker(Δ)/im(Norm)] /
#[ker(Norm)/im(Δ)] =
= # [U_{k}/Norm_{Nk}U_{N}] /
# [E_{Nk}/U_{N}^{1  σ}],
where Δ : E → E^{1  σ} and
Norm : E → E^{1 + σ+ … + σ4}
are endomorphisms of U_{N}
and E_{Nk} consists of the units E in U_{N}
with relative norm Norm_{Nk}(E) = 1.
According to Takagi (1920), Hasse (1927) and Herbrand (1932),
the Herbrand quotient of the Gal(Nk)module U_{N} has the value
#H^{0}(Gal(Nk),U_{N}) /
#H^{1}(Gal(Nk),U_{N}) =
1 / [N : k],
since Archimedean places do not yield a contribution.
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over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):
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