Journal of Number Theory 130 (2010) 304–306

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Journal of Number Theory

www.elsevier.com/locate/jnt

Remark on infinite unramified extensions of number fieldswith class number one

David Brink ∗

Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100, Denmark

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 November 2008Revised 10 August 2009Available online 24 October 2009Communicated by Michael E. Pohst

MSC:12F10

Keywords:Non-solvable unramified extensions

We modify an idea of Maire to construct biquadratic numberfields with small root discriminants, class number one, and havingan infinite, necessarily non-solvable, strictly unramified Galoisextension.

© 2009 Elsevier Inc. All rights reserved.

Let k be an algebraic number field with class number one. Then k has no Abelian (and hence nosolvable) non-trivial unramified Galois extension. It is somewhat surprising that k may neverthelesshave a non-solvable unramified extension. Many such examples are known, cf. [9]. The followingexample 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 onehaving an infinite unramified extension. It is the purpose of this note to show how Maire’s ingeniousmethod can be modified in order to find other such examples, but with considerably smaller rootdiscriminants.

Theorem. Assume first that f ∈ Z[x] is an irreducible polynomial of degree five with only real roots andwhose discriminant l is a prime such that Q(

√l ) has class number one. Assume further that q1 and q2 are

primes such that Q(√

q1q2 ) has class number one and Q(√

lq1q2 ) has class number two. Assume finally thatf has five simple roots modulo q1 , and that the tuple μ of the degrees of the irreducible factors of f modulo q2

* Address for correspondence: University College Dublin, School of Mathematical Sciences, Belfield, Dublin 4, Ireland.E-mail address: brink@math.ku.dk.

0022-314X/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.jnt.2009.08.013

D. Brink / Journal of Number Theory 130 (2010) 304–306 305

is (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 andan infinite unramified extension.

Proof. (Cf. [4].) It follows from the first assumption by a result of Kondo [3] that the splitting field Kof f is an S5-extension of Q and 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 k has class number one.Let r be the number of primes p in K ramified in M . It is a result of Martinet [6] that M has

infinite 2-class field tower if

r � [K : Q] + 3 + 2√[M : Q] + 1 = 123 + 2

√241 ≈ 154.

Let θ be a root of f . It follows from the third assumption that q1 splits completely in Q(θ)

and hence in K too, and that q2 decomposes in Q(θ) as q2 = p1 · · ·pr with inertia degrees(deg(p1), . . . ,deg(pr)) = μ. Let ZP ⊆ Gal(K/Q) = S5 be the decomposition group of some prime P

in K dividing q2. It is cyclic since q2 is unramified. By a result of Artin (see [8]), the cycle type of agenerator of ZP equals μ. Hence ZP has order at most three. It now follows that K has 120 primesdividing q1 and at least 40 primes dividing q2. Then r � 120 + 40 since they all ramify in M , and theclaim 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 herealong with suitable primes qi found with the aid of the computer program PARI:

f l q1 q2

x5 − 2x4 − 3x3 + 5x2 + x − 1 36 497 2819 103x5 − x4 − 5x3 + 3x2 + 5x − 2 81 509 1123 47

We conclude from the theorem that both fields Q(√

36 497,√

2819 · 103 ) and Q(√

81 509,√

1123 · 47 )

have class number one and an infinite unramified extension. Of these, the second has the smallerroot discriminant, namely about 65 591. The corresponding root discriminants in [4] are all greaterthan 1011.

We conclude this note by discussing the possibility of finding similar examples with other typesof base fields. Any number field having an infinite unramified extension has root discriminant atleast 4πeγ ≈ 22.4 (resp. 8πeγ ≈ 44.8 under GRH) by results of Odlyzko and Serre (see the dis-cussion in [5]). Hence none of the nine imaginary quadratic fields with class number one has aninfinite unramified extension (in fact, none of them has any non-trivial unramified extensions atall [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 k hasno such extension using merely the magnitude of its discriminant.

Finally consider real quadratic fields. Using the same arguments as above, one can show the fol-lowing:

Assume that f is an irreducible polynomial of degree 20 with only real roots and whose discriminant is ofthe form q1q2 with primes q1 and q2 . Assume further that f has 18 simple roots and one double root moduloboth q1 and 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 arandom polynomial of degree 20 has 18 simple roots modulo qi is very small.

306 D. Brink / Journal of Number Theory 130 (2010) 304–306

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, in: European Congress of Mathematics, vol. II, Barcelona,2000, in: Progr. Math., vol. 202, Birkhäuser, Basel, 2001, pp. 207–218.

[3] T. Kondo, Algebraic number fields with the discriminant equal to that of a quadratic number field, J. Math. Soc. Japan 47 (1)(1995) 31–36.

[4] C. Maire, On infinite unramified extensions, Pacific J. Math. 192 (1) (2000) 135–142.[5] J. Martinet, Petits discriminants des corps de nombres, Lecture Notes London Math. Soc. 56 (1982) 151–193.[6] J. Martinet, Tours de corps de classes et estimations de discriminants, Invent. 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 Trägheitsgruppe als Permutationsgruppen, Math. Ann. 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. Théor. NombresBordeaux 9 (1997) 405–448.