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Remark on infinite unramified extensions of number fields with class number one

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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.
Remark on infinite unramified extensions of
number fields with class number one
David Brink
July 2009
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. [9]. 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 x711x5+ 17x35x+ 1. Here and in
what follows, we always understand the word “unramified” in the strict sense.
Recently, Maire [4] 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 fZ[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. [4]). It follows from the first assumption by a result of Kondo [3] 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 [4] that khas class
number one.
1
Let rbe the number of primes pin Kramified in M. It is a result of Martinet
[6] that Mhas infinite 2-class field tower if
r[K:Q]+3+2p[M:Q] + 1 = 123 + 2241 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 ZPGal(K/Q) = S5be
the decomposition group of some prime Pin Kdividing q2. It is cyclic since
q2is unramified. By a result of Artin (see [8]), 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 r120 + 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 [9]. We present them here along with suitable primes qifound with
the aid of the computer program PARI:
f l q1q2
x52x43x3+ 5x2+x1 36497 2819 103
x5x45x3+ 3x2+ 5x2 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 [4] 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 4πeγ22.4 (resp. 8πeγ44.8 under
GRH) by results of Odlyzko and Serre (see the discussion in [5]). 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 [10]).
There are 47 imaginary biquadratic number fields with class number one [1].
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 [2], one can never show that khas no such extension using
merely the magnitude of its discriminant.
2
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.
References
[1] E. Brown, C. J. Parry, The imaginary bicyclic biquadratic fields with class
number 1, J. Reine Angew. Math. 266 (1974), 118–120.
[2] F. Hajir, C. Maire, Asymptotically good towers of global fields, European
Congress of Mathematics, Vol. II (Barcelona, 2000), 207–218, Progr. Math.,
202, Birkh¨auser, Basel, 2001.
[3] T. Kondo, Algebraic number fields with the discriminant equal to that of a
quadratic number field, J. Math. Soc. Japan 47 (1995), no. 1, 31–36.
[4] C. Maire On infinite unramified extensions, Pacific J. Math. 192 (2000),
no. 1, 135–142.
[5] J. Martinet, Petits discriminants des corps de nombres, Lecture Notes of the
London Math. Soc. 56 (1982), 151–193.
[6] J. Martinet, Tours de corps de classes et estimations de discriminants, Inv.
Math. 44 (1978), 65–73.
[7] M. Pohst, H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge
University Press, Cambridge, 1989.
[8] B. L. van der Waerden, Die Zerlegungs- und Tagheitsgruppe als Permuta-
tionsgruppen, Math. Annalen 111 (1935), 731–733.
[9] K. Yamamura On unramified Galois extensions of real quadratic number
fields, Osaka J. Math. 23 (1986), 471–478.
[10] K. Yamamura Maximal unramified extensions of imaginary quadratic number
fields of small conductors, J. Th´eor. Nombres Bordeaux 9(1997), 405–448.
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