Remark on inﬁnite unramiﬁed extensions of
number ﬁelds with class number one
Abstract. We modify an idea of Maire to construct biquadratic number
ﬁelds with small root discriminants, class number one, and having an
inﬁnite, necessarily non-solvable, strictly unramiﬁed Galois extension.
Let kbe an algebraic number ﬁeld with class number one. Then khas no Abelian
(and hence no solvable) non-trivial unramiﬁed Galois extension. It is somewhat
surprising that kmay nevertheless have a non-solvable unramiﬁed extension.
Many such examples are known, cf. . The following example is perhaps new:
The ﬁeld k=Q(√29,√4967) has class number one and an unramiﬁed PSL(2,7)-
extension given as the splitting ﬁeld of x7−11x5+ 17x3−5x+ 1. Here and in
what follows, we always understand the word “unramiﬁed” in the strict sense.
Recently, Maire  showed that there are even biquadratic number ﬁelds with
class number one having an inﬁnite unramiﬁed extension. It is the purpose of
this note to show how Maire’s ingenious method can be modiﬁed in order to ﬁnd
other such examples, but with considerably smaller root discriminants.
Theorem. Assume ﬁrst that f∈Z[x]is an irreducible polynomial of degree ﬁve
with only real roots and whose discriminant lis a prime such that Q(√l)has class
number one. Assume further that q1and q2are primes such that Q(√q1q2)has
class number one and Q(√lq1q2)has class number two. Assume ﬁnally that fhas
ﬁve simple roots modulo q1, and that the tuple µof the degrees of the irreducible
factors of fmodulo q2is (1,1,1,1,1), (1,1,1,2), (1,2,2) or (1,1,3). Then the ﬁeld
k=Q(√l, √q1q2)has class number one and an inﬁnite unramiﬁed extension.
Proof (cf. ). It follows from the ﬁrst assumption by a result of Kondo  that
the splitting ﬁeld Kof fis an S5-extension of Qand an unramiﬁed A5-extension
of Q(√l). Hence M=K(√q1q2) is an unramiﬁed A5-extension of k.
It follows from the second assumption by the argument in  that khas class
Let rbe the number of primes pin Kramiﬁed in M. It is a result of Martinet
 that Mhas inﬁnite 2-class ﬁeld tower if
r≥[K:Q]+3+2p[M:Q] + 1 = 123 + 2√241 ≈154.
Let θbe a root of f. It follows from the third assumption that q1splits completely
in Q(θ) and hence in Ktoo, and that q2decomposes in Q(θ) as q2=p1···pr
with inertia degrees (deg(p1),...,deg(pr)) = µ. Let ZP⊆Gal(K/Q) = S5be
the decomposition group of some prime Pin Kdividing q2. It is cyclic since
q2is unramiﬁed. By a result of Artin (see ), the cycle type of a generator of
ZPequals µ. Hence ZPhas order at most three. It now follows that Khas 120
primes dividing q1and at least 40 primes dividing q2. Then r≥120 + 40 since
they all ramify in M, and the claim follows.
There are only two totally real quintic ﬁelds with prime discriminant l < 100,000
such that Q(√l) has class number one [7, p. 442]. These were also studied by
Yamamura . We present them here along with suitable primes qifound with
the aid of the computer program PARI:
f l q1q2
x5−2x4−3x3+ 5x2+x−1 36497 2819 103
x5−x4−5x3+ 3x2+ 5x−2 81509 1123 47
We conclude from the theorem that both ﬁelds Q(√36497,√2819 ·103) and
Q(√81509,√1123 ·47) have class number one and an inﬁnite unramiﬁed exten-
sion. Of these, the second has the smaller root discriminant, namely about 65591.
The corresponding root discriminants in  are all greater than 1011 .
We conclude this note by discussing the possibility of ﬁnding similar examples
with other types of base ﬁelds. Any number ﬁeld having an inﬁnite unramiﬁed
extension has root discriminant at least 4πeγ≈22.4 (resp. 8πeγ≈44.8 under
GRH) by results of Odlyzko and Serre (see the discussion in ). Hence none of
the nine imaginary quadratic ﬁelds with class number one has an inﬁnite unram-
iﬁed extension (in fact, none of them has any non-trivial unramiﬁed extensions
at all ).
There are 47 imaginary biquadratic number ﬁelds with class number one .
Of these, 37 (resp. 43) have root discriminants less than 4πeγ(resp. 8πeγ). On the
negative side, k=Q(√−67,√−163) has root discriminant about 104.5. Since
there exists a number ﬁeld with root discriminant about 84.4 having an inﬁnite
unramiﬁed extension , one can never show that khas no such extension using
merely the magnitude of its discriminant.
Finally consider real quadratic ﬁelds. Using the same arguments as above,
one can show the following:
Assume that fis an irreducible polynomial of degree 20 with only real roots and
whose discriminant is of the form q1q2with primes q1and q2. Assume further
that fhas 18 simple roots and one double root modulo both q1and q2. Then
k=Q(√q1q2)has an inﬁnite unramiﬁed extension.
Finding a suitable polynomial f, however, seems to be very diﬃcult, since the
probability that a random polynomial of degree 20 has 18 simple roots modulo
qiis very small.
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