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Pure Quintic Fields
§ 1. Galois closure and subfield lattice.
Let q_{1},…,q_{s} be pairwise distinct primes
such that s ≥ 1 and 5 may be among them.
Denote by L = Q(D^{1/5}) the pure quintic number field
with fifth power free radicand D = q_{1}^{e1} … q_{s}^{es},
where the exponents are integers 1 ≤ e_{j} ≤ 4.
The field L is generated by adjoining the unique real solution
of the pure quintic equation X^{5}  D = 0 to the rational number field Q.
It is a nonGalois algebraic number field with signature (1,2)
and thus possesses four isomorphic complex fields
L_{j} = Q(ζ^{j}D^{1/5}),
1 ≤ j ≤ 4,
all of whose arithmetical invariants coincide.
The normal closure of L is the compositum N = Q(D^{1/5},ζ)
of L = Q(D^{1/5}) with the cyclotomic field k = Q(ζ),
where ζ = ζ_{5} = exp(2 π i / 5) denotes a primitive fifth root of unity.
N is a complex metacyclic field of degree 20 whose Galois group
Gal(NQ)
is the semidirect product M_{5} = C_{5} * C_{4} of two cyclic groups.
The cyclotomic field k is a complex cyclic quartic field
and contains the real quadratic field Q(5^{1/2})
as its maximal real subfield k^{+} = Q(ζ + ζ^{1}).
Consequently, there exists a real intermediate field
M = Q(D^{1/5},5^{1/2}) of degree 10 between L and N,
which is nonGalois with signature (2,4).
§ 2. Class numbers and unit index.
In 1973, Parry has determined the class number relation
5^{5} h_{N} = (U_{N}:U_{0}) h_{L}^{4}
[Thm. I, p. 476, Pa],
where
U_{N} denotes the unit group of N,
U_{0} is the subgroup of U_{N} generated by
all units of the conjugate fields
L_{j} = Q(ζ^{j}D^{1/5}),
0 ≤ j ≤ 4,
of L and of k,
and the unit index (U_{N}:U_{0}) can take
seven possible values 5^{e} with 0 ≤ e ≤ 6
[Thm. II, p. 478, Pa].
Examples 2.1.
Anticipating some of our numerical results in section 8,
we give the smallest radicands D of pure quintic fields L = Q(D^{1/5})
where the various values of the exponent e actually occur:
e = 6 for D = 6 = 2*3 of type γ,
e = 5 for D = 2 of type ε,
e = 4 for D = 22 = 2*11 of type β_{2},
e = 3 for D = 11 of type α_{2},
e = 2 for D = 31 of type α_{1},
e = 1 for D = 33 = 3*11 of type α_{2}.
However, we point out that we did not find a realization of
e = 0,
that is, U_{N} is never generated completely by subfield units.
At the end of his article, Parry suggests verbatim
"In conclusion the author would like to say that he believes
a numerical study of pure quintic fields would be most interesting"
[p. 484, Pa].
Of course, it would have been rather difficult
to realize Parry's desire in 1973.
But now, 40 years later, we are in the position to use
the powerful computer algebra systems PARI/GP and MAGMA
to start an attack against this hard problem.
This will be done in our present research line.
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Trade, Science, Art and Industry
Principal investigator of the
International Research Project
Towers of pClass Fields
over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):
P 26008N25
