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Pure Quintic Fields 1
§ 3. Conductor and discriminants.
Let R = q1 … qs be the squarefree product
of all prime divisors of the radicand D
of the pure quintic field L = Q(D1/5).
Independently of the exponents e1,…,es,
the conductor f of the cyclic quintic relative extension N|k
is given by
f4 = 52 R4
if not D ≡ ±1,±7 (mod 25) (field of the first kind),
f4 = R4
if D ≡ ±1,±7 (mod 25) (field of the second kind)
[Thm. 1, p. 103, Ma1].
The smallest radicands D of pure quintic fields L = Q(D1/5)
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 5|D,
D = 7, 18 = 2*32, 126 = 2*7*32, for fields of the second kind.
It is well known that the cyclotomic discriminant takes the value
d(k) = +53 = 125,
and Hilbert's Theorem 39 on discriminants of composite fields shows the following
The metacyclic discriminant is given by
d(N) = d(k)5 f16 =
= 523 R16 for a field of the first kind,
= 515 R16 for a field of the second kind,
the intermediate discriminant is given by
d(M) = 5 d(k)2 f8 =
= 511 R8 for a field of the first kind,
= 57 R8 for a field of the second kind,
and the pure quintic discriminant is given by
d(L) = d(k) f4 =
= 55 R4 for a field of the first kind,
= 53 R4 for a field of the second kind.
§ 4. Herbrand quotient of units of N|k.
In section 2, we have seen that the unit index
(UN:U0) = 5e
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 UN
with respect to the cyclic quintic Kummer extension N|k
provides additional structural information.
The Herbrand quotient of UN under the action of Gal(N|k) = < σ >
is defined by
= #[ker(Δ)/im(Norm)] /
= # [Uk/NormN|kUN] /
# [EN|k/UN1 - σ],
where Δ : E → E1 - σ and
Norm : E → E1 + σ+ … + σ4
are endomorphisms of UN
and EN|k consists of the units E in UN
with relative norm NormN|k(E) = 1.
According to Takagi (1920), Hasse (1927) and Herbrand (1932),
the Herbrand quotient of the Gal(N|k)-module UN has the value
1 / [N : k],
since Archimedean places do not yield a contribution.
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