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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 x7−11x5+ 17x3−5x+ 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 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. [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 + 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 [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 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 [9]. 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 [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 4πeγ≈22.4 (resp. 8πeγ≈44.8 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 Tr¨agheitsgruppe 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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