MINIMUM DISCRIMINANTS OF IMPRIMITIVE DECICFIELDSERIC D. DRIVER AND JOHN W. JONES
Abstract. In this paper, we compute the minimum discriminants ofimprimitive degree 10 fields for different combinations of Galois groupand signature. We use class field theory when there is a quintic subfield,
and a Martinet search in the more difficult case where there is only aquadratic subfield.

If L is a degree n number field, let DL be its discriminant, and dL := |DL |its absolute discriminant. Let G(L) be a transitive subgroup of Sn giving
the Galois group of the Galois closure of L/Q, which is well-defined up toconjugation. We let r1 denote the number of real places of L.
For a given positive integer n and group G ≤ Sn , several authors havecontributed to our knowledge of minimizing dL for fields L with [L : Q] = nand G(L) = G. In [Klüners and Malle 2001], Klüners and Malle discuss themore refined problem of minimizing dL with n, G, and r1 fixed, as well as
the still more refined analogue of this question where the conjugacy class ofcomplex conjugation specified. Their results are primarily for degrees ≤ 8,
and there have been some results in degree 9. Here we consider imprimitive
fields in degree n = 10 and attempt to determine minimal examples for fixed
G and r1 .
The computation for imprimitive decics divides naturally into three cases,
which we label as follows. We refer to decics L which contain a quadratic
but not quintic subfield as Type 2 fields, those which contain a quintic
but no quadratic subfield as Type 5 fields, and those which contain bothas Type 2-5 fields. The most difficult cases are Type 2 fields where weemployed computer searches based on Martinet’s generalization of Hunter’stheorem. Some of these were standard Martinet searches, and some weretargeted Martinet searches in the sense of [Driver and Jones 2009]. This is
explained in Section 1.
Section 2 treats Type 5 decics using class field theory. These computations
rely on having a sufficiently large complete list of quintic fields to workfrom. We extended the range of known totally real quintic fields K fromdK ≤ 2 · 107 to dK ≤ 108 which in turn, allowed us to determine moreminimal decic discriminants.
Finally, in Section 3 we describe results for the simplest case, Type 25 decic fields. These fields are just tensor products of their subfields, so1

2ERIC D. DRIVER AND JOHN W. JONES

working with tables of lower degree fields suffices here. Again, we employed
our extended list of totally real quintic fields here.
1. Type 2 Decics
1.1. The Martinet Search. A standard technique for searching for all
fields with absolute discriminant less than a given bound comes from thegeometry of numbers. In particular, all such primitive degree n extensionsof a degree m field can in principle be found using Martinet’s theorem, ageneralization of Hunter’s theorem [Cohen 2000, Sec 9.3].
Here, we use Martinet’s theorem to find all quintic extensions of a quadratic base field. If a decic field L with a quadratic subfield K satisfies thediscriminant bound dL ≤ B, then the discriminant of K must satisfy
1/5

dK ≤ dL ≤ B 1/5 .
In this paper, we take B = 1010 . Thus, it is necessary to search over eachquadratic base field satisfying dK ≤ 100; there are 61 such quadratic fields.
In [Driver and Jones 2009], we show how to target particular discriminants in a Martinet search. Note, our search here is naturally broken up
into 61 searches, one for each quadratic field. When the quadratic base field
has relatively large discriminant there are correspondingly few possibilities
for the decic discriminant beneath√ the bound B. For example, when thequadratic base field is K = Q( 97), the only possibility for the decic fieldis dL = 975 .
When we have a priori knowledge of the absolute decic discriminant, wecan first test to make sure that the ratio of the absolute polynomial discriminant to dL is a square. This usually weeds out 99% of the candidate
polynomials, and allows us to skip the more computationally expensive tasksof checking for irreducibility and computing field discriminants. Even whenmultiple discriminants are possible, it is usually much faster to use several
targeted searches instead of a single standard search.
The use of targeting is only helpful here for those base fields having thelargest discriminant bounds; as the discriminant of the base field drops, apoint is reached where too much targeting would be required in order to bepractical. In our implementation, we found that a good rule of thumb is touse a targeted Martinet search for those base fields having dK ≥ .75B 1/5 .
1.2. Results. For our first attempt, we used a discriminant bound of B =109 . The Martinet search took approximately one week on a 3GHz Pentium,and found 239 fields.
To extend the search further, we used a distributed computing approachthrough the BOINC system [BOINC]. With B = 1010 , the computation tooka total of approximately 20,000 machine hours (summed over all hosts). It
found 10,565 fields. Table 1 shows the distribution of fields partitioned bytype and signature. If no fields of a particular Galois group (resp. signature)

MINIMUM DISCRIMINANTS OF IMPRIMITIVE DECIC FIELDS

3

Table 1: Numbers of decics L with a quadratic subfield and dL ≤ 1010 .

Signature
Gal (0,5) (2,4) (4,3) Total
T1
T2
T3
T5
T6
T 11
T 21
T 22T 33T 40T 41
T 43

12616214158
1259187

2001214040946

202

12816226172
16510335

were found then the corresponding row (resp. column) was left out of thetable. Impossible combinations of type and signature are grayed out.
Table 2 gives the minimum discriminants for Type 2 decics as determined
Table 2: Minimum discriminants and defining polynomials of Type 2 decics.

Gal r1Discrim.
T6 0−35 314
T 21 0−210 7612
T 21 255 6012
T 33 0−35 3373
T 40 0−35 49032
T 40 255 14292
T 41 0−26 35 132 432
T 43 0−35 112 7369
T 43 255 280001
T 43 4 −55 10391 1049

Polynomialx10 + 2x8 − 3x7 + 3x6 − 7x5 + 8x4 − 7x3 + 7x2 − 4x + 1x10 − 4x9 + 8x8 − 10x7 + 9x6 − 6x5 + 3x4 − x2 + 1x10 − 3x9 + 5x8 − 5x7 + 2x6 + 2x5 − 4x4 + 2x3 − 2x + 1x10 − 4x9 + 7x8 − 11x7 + 18x6 − 19x5 + 16x4 − 17x3 + 10x2 − 3x + 3x10 − x9 + 5x8 − 3x7 + 8x6 − 4x5 + 4x4 − 3x3 − x + 1x10 − 4x9 + 8x8 − 12x7 + 18x6 − 21x5 + 18x4 − 9x3 + x2 + 2x − 1x10 − x9 − 4x8 + x7 + 8x6 + 2x5 − 6x4 − x3 + 7x2 + 5x + 1
x10 − 3x9 + 5x8 − 6x7 + 6x6 − 4x5 + 2x4 − x3 + x2 − x + 1
x10 − x9 − x8 + 3x7 − 3x6 − x5 + 5x4 − x3 − 3x2 + x + 1
x10 − 3x9 + 5x8 − 4x7 − 2x6 + 8x5 − 8x4 + x3 + 3x2 − 3x + 1

by our computer search. Since these polynomials were computationally themost difficult to obtain, we include defining polynomials of the corresponding

4ERIC D. DRIVER AND JOHN W. JONES

fields. Here, signatures are given by just r1 , the number of real places.
Discriminants are given in factored form. Here, and in subsequent tables,
we include exponents of 1 in a factorization if needed to separate prime
factors.
2. Type 5 Decics
2.1. Totally Real Quintics. Studying Type 5 decics relies on our knowledge of quintic fields. In the case of totally real quintics K, those withdK ≤ 2 · 107 were classified by [Schwarz, Pohst, and Diaz Y Diaz 1994], andare available at [Bordeaux]. While the bound for dK is large compared tothe corresponding bounds for the two other signatures of quintic fields, it
only accounts for 22,740 fields which we found insufficient.
We implemented a standard Hunter’s search using the methods describedin [Cohen 2000, §9.3] to find all totally real quintic fields K with dK ≤ 108 .
The number of fields broken down by Galois group is given in Table 3.
Here we use common names for the Galois groups, which correspond toTable 3: Numbers of totally real quintic number fields K with dK ≤ 108 ,partitioned by Galois group.
Gal C5 D5 F5 A5S5
#6 72 59 80 162,022
T1–T5 respectively using the labelling of [Butler and McKay 1983]. Thefields themselves are available at [Jones and Roberts]. As was the case in[Schwarz, Pohst, and Diaz Y Diaz 1994], almost all of the fields have Galois
group S5 . Although this is a fairly modest increase in the range of discriminants, it was of great use in computing the minimal absolute discriminants
totally real decics of Type 5 and Type 2-5.
2.2. Class Field Theory. For decic fields containing a quintic subfield,
class field theory provides a mechanism for computing the desired fields.
Here, computations are relatively simple in comparison to more general casesof Kummer theory since our fields always contain ζ2 = −1, so the desired
extensions can be found by taking square roots of appropriate elements. An
algorithm for this is given in [Cohen 2000, §9.2.2].
All computations here were carried out with gp [PARI2]. We needed todeviate from the approach given in [Cohen 2000] because of memory issues.
Given a base field K and bound B, a quadratic extension corresponding
to modulus (m0 , m∞ ) needs to satisfy d2K N (m0 ) ≤ B. In some cases, itwas impractical to first compute all m0 with N (m0 ) ≤ B/d2K . Instead, we
compute theq list L of all such moduli which are only divisible by primesp ≤ B1 :=B/d2K . All additional moduli are then of the form m0 Q whereMINIMUM DISCRIMINANTS OF IMPRIMITIVE DECIC FIELDS

5Q divides a prime q with B1 < q ≤ B/d2K , and m0 from L. So, as wecompute these, we compute the corresponding fields as we go and do nothave to store these additional moduli.
For example, consider the totally real S5 quintic field of minimal discriminant. It has discriminant 61 · 397. With B = 1016 , the first phase foundand stored the 766,157 moduli which are divisible only by primes less thanB1 = 4129. The total number of moduli was 3,851,600.
One could compute decic overfields of quintics using a Martinet search,as was done in [Selmane 2000a, Selmane 2000b, Selmane 2002]. We found,however, that class field theory was much more efficient.
2.3. Results. For complete searches of Type 5 decics, we computed all fields
L with dL ≤ 1012 . When the quintic subfield was totally real, we computed
all decics with dL ≤ 1013 . The results are summarized in Table 4. These
Table 4: Numbers of Type 5 decic fields L with dL ≤ B by signature.
Columns correspond to the number of real places for the decic field, androws correspond to the number of real places of the quintic subfield.
r1B

13510121012
1013

0246810226764 210453
11972 37249 32852 7876
7723954 7527 7592 2849 313

computations include, and extend, those performed in [Selmane 2000a, Selmane 2000b,Selmane 2002]. We found one discrepancy with these prior computations,
namely we found one more totally complex decic field with a quintic subfield with r1 = 3 and dL ≤ 1011 . Otherwise, our results are consistent with
[Selmane 2000a, Selmane 2000b, Selmane 2002].
Table 5 gives our findings for minimal absolute discriminants for Type 5decics. Most entries come from the searches summarized above, namelyfrom fields with dL ≤ 1012 . Almost all of the cases not settled by thatsearch had a totally real quintic subfield. This naturally includes totallyreal decics (r1 = 10), but also cases where (T, r1 ) = (16, 2), (16, 6), (24, 6),(25, 6), (29, 4), (29, 6), (29, 8), (34, 6), (36, 4), (36, 6), (36, 8), (38, 6).
For example, consider a T 16 decic L with r2 = 2. The group T 16 has twoconjugacy classes of elements of order 2 and cycle type 22 16 . The uniquequintic subfield K has G(K) = D5 , and in the projection T 16 → D5 inducedby the inclusion K ,→ L, elements of cycle type 22 16 map to the identity.
So, for a T 16 decic with r1 = 2, the quintic subfield is a totally real D5 field.
In these cases, we made use of the quintics computed in Section 2.1. In
each case, we computed decic overfields of the desired signature, where thequintic had a the correct Galois group, and dL ≤ 1016 . In this computation,

6

ERIC D. DRIVER AND JOHN W. JONES

Table 5: Minimum discriminants for Type 5 decics.

Gal(0, 5)(2, 4)

Signature(4, 3)(6, 2)

118 232

118 232(8, 1)(10, 0)210 118 232

T8
T 14 −118 1451
118 89−118 23
118 109−118 331
118 431 67
10424
T 15
2 47
3 401672 4104
T 16−52 475
32 40154015
172 4015
T 23−474 83
474 193−31 4014
131 4014−33 4014
3971 4014
T 24
216 136
210 510 74?
127T 25
2 13210 511 74?
8686816816816810T 29 −2 13 31
2 13 41−2 3 53
2 5 53−2 19 53
2 5 74 97
T 34?
34 8834
26 114 1134
1241414414614544
T 36 −2 17 43 2 11 17−2 3 881
2 17 881−3 883
2 114 291 1134
T 37
52 57832
210 132 3472
210 364972
T 38−38 2233
28 2273−28 134 193
613 3973
24 337693
T 39 −72 231 4312 72 411 6312 −72 431 12172 4611 57832 −612 1671 3972 612 3972 757
we only kept track of fields of minimal discriminant for each Galois groupsince the number of fields in some cases was so large.
3. Type 2-5 Decics
The simplest case is where the decic contains both a quadratic and quintic
subfield. There are 8 corresponding decic Galois groups. These fields are easily generated from tables of quadratics and quintics by forming composita.
We used quintics from the Bordeaux ftp site [Bordeaux], augmented by theresults of our search for totally real quintics described in Section 2.1 above.
Naturally, some care must be exercised to insure that one has searched farenough. Using this method, we were able to find the minimum discriminants
for all the above mentioned decics, and for almost all possible signatures,with the results given in Table 6.
References[BOINC] BOINC, Berkeley Open Infrastructure for Network Computing. http://boinc.
berkeley.edu[Bordeaux] The Bordeaux Database of Number Fields, ftp://megrez.math.u-bordeaux.
fr/pub/numberfields[Butler and McKay 1983] G. Butler and J. McKay, The transitive groups of degree up to11, Comm. Algebra 11 (1983), 863–911.
[Cohen 2000] H. Cohen, Advanced Topics in Computational Number Theory, SpringerVerlag, New York, 2000.MINIMUM DISCRIMINANTS OF IMPRIMITIVE DECIC FIELDS

7Table 6: Minimum discriminants of Type 2–5 decics.

Gal(0, 5)Signature(2, 4)(6, 2)(10, 0)−119

T1
35 119T2−4754015
54
45T3−3 47
7 1755 4014
T428 511
28 511 74
T5−35 176
215 510215 74 136
T 11 −216 174
55 53438 55 734
10488
7
T 12 −2 89
2 3 528 137 174
T 22 −35 11232 55 132 412 135 3472 55 612 3972
[Driver and Jones 2009] E. Driver and J. Jones, A targeted Martinet search, Math. Comp.
78 (2009), 1109–1117.
[Jones and Roberts] J. Jones and D. Roberts, Tables of number fields with prescribed
ramification, http://math.la.asu.edu/~jj/numberfields[Klüners and Malle 2001] J. Klüners and G. Malle, A database for field extensions of therationals, LMS J. Comput. Math. 4 (2001), 182–196.
[PARI2] PARI/GP, version 2.3.3, Bordeaux, 2008, http://pari.math.u-bordeaux.fr/.
[Schwarz, Pohst, and Diaz Y Diaz 1994] A. Schwarz, M. Pohst, and F. Diaz Y Diaz, A
table of quintic number fields, Math. Comp. 63 (1994), 361–376.
[Selmane 2000a] S. Selmane, Quadratic extensions of totally real quintic fields, Math.
Comp. 70 (2000), 837–843.
[Selmane 2000b] S. Selmane, Tenth degree number fields with quintic fields having one
real place, Math. Comp. 70 (2000), 845–851.
[Selmane 2002] S. Selmane, Quadratic extensions of quintic fields of signature (3, 1), Acta
Math. Inform. Univ. Ostraviensis 10(1) (2002), 117–123.
Lockheed Martin, 4318 N. 69th Place, Scottsdale, 85251
E-mail address: eric.driver@lmco.com
School of Mathematical and Statistical Sciences, Arizona State University, Box 871804, Tempe, AZ 85287
E-mail address: jj@asu.edu