Minimal discriminants of number fields by degree and signature

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Numbers

$n$

$r_2$

$d_K$

comment: $x^2-x-1=0$; Galois group $C_2$ (2T1); $h_K=1$; LMFDB 2.2.5.1; $K=\mathbb{Q}(\sqrt5)$; the discriminant of a quadratic field is a fundamental discriminant, and $5$ is the least positive one; the next field of signature $(2,0)$ has $|d_K|=8$

-3

comment: $x^2-x+1=0$; Galois group $C_2$ (2T1); $h_K=1$; LMFDB 2.0.3.1; $K=\mathbb{Q}(\sqrt{-3})=\mathbb{Q}(\zeta_3)$; the discriminant of a quadratic field is a fundamental discriminant, and $-3$ is the negative one of least absolute value; the next field of signature $(0,1)$ has $|d_K|=4$

comment: $x^3-x^2-2x+1=0$; Galois group $C_3$ (3T1); $h_K=1$; LMFDB 3.3.49.1; $K=\mathbb{Q}(\zeta_7)^+$, the maximal real subfield of the seventh cyclotomic field; the minimum follows from the list of every cubic field with $|D|\leq 3000$ in the table of regulators of cubic fields, which rests on Hunter's theorem [18]; the next field of signature $(3,0)$ has $|d_K|=81$

-23

comment: $x^3-x^2+1=0$; Galois group $S_3$ (3T2); $h_K=1$; LMFDB 3.1.23.1; $K=\mathbb{Q}(\rho)$ for the plastic number $\rho$, the real root of $x^3-x-1$; the minimum follows from the list of every cubic field with $|D|\leq 3000$ in the table of regulators of cubic fields, which rests on Hunter's theorem [18]; the next field of signature $(1,1)$ has $|d_K|=31$

725

comment: $x^4-x^3-3x^2+x+1=0$; Galois group the dihedral group of order 8 (4T3); $h_K=1$; LMFDB 4.4.725.1; quadratic subfield $\mathbb{Q}(\sqrt{5})$; the minimum follows from the complete tables of quartic fields with $|d_K|<10^6$ [1] [2]; the next field of signature $(4,0)$ has $|d_K|=1125$

-275

comment: $x^4-x^3+2x-1=0$; Galois group the dihedral group of order 8 (4T3); $h_K=1$; LMFDB 4.2.275.1; quadratic subfield $\mathbb{Q}(\sqrt{5})$; the minimum follows from the complete table of quartic fields with $|d_K|<10^6$ [2]; the next field of signature $(2,1)$ has $|d_K|=283$

117

comment: $x^4-x^3-x^2+x+1=0$; Galois group the dihedral group of order 8 (4T3); $h_K=1$; LMFDB 4.0.117.1; quadratic subfield $\mathbb{Q}(\sqrt{-3})$; the minimum follows from the complete table of quartic fields with $|d_K|<10^6$ [2]; the next field of signature $(0,2)$ has $|d_K|=125$

14641

comment: $x^5-x^4-4x^3+3x^2+3x-1=0$; Galois group $C_5$ (5T1); $h_K=1$; LMFDB 5.5.14641.1; $K=\mathbb{Q}(\zeta_{11})^+$, the maximal real subfield of the eleventh cyclotomic field, $d_K=11^4$; the minimum follows from the complete tables of quintic fields [3] [4]; the next field of signature $(5,0)$ has $|d_K|=24217$

-4511

comment: $x^5-x^3-2x^2+1=0$; Galois group $S_5$ (5T5); $h_K=1$; LMFDB 5.3.4511.1; the minimum follows from the complete table of quintic fields [4]; the next field of signature $(3,1)$ has $|d_K|=4903$

1609

comment: $x^5-x^3-x^2+x+1=0$; Galois group $S_5$ (5T5); $h_K=1$; LMFDB 5.1.1609.1; the minimum follows from the complete table of quintic fields [4]; the next field of signature $(1,2)$ has $|d_K|=1649$

300125

comment: $x^6-x^5-7x^4+2x^3+7x^2-2x-1=0$; Galois group $C_6$ (6T1); $h_K=1$; LMFDB 6.6.300125.1; quadratic subfield $\mathbb{Q}(\sqrt{5})$; $K=\mathbb{Q}(\sqrt5)\cdot\mathbb{Q}(\zeta_7)^+$, the cyclic sextic field of conductor $35$, $d_K=5^3\cdot7^4$; the minimum was proved by Pohst [6]; the next field of signature $(6,0)$ has $|d_K|=371293$

-92779

comment: $x^6-x^5-2x^4+3x^3-x^2-2x+1=0$; Galois group $S_6$ (6T16); $h_K=1$; LMFDB 6.4.92779.1; the minimum was proved by Pohst [6]; the next field of signature $(4,1)$ has $|d_K|=94363$

28037

comment: $x^6-2x^5+3x^3-2x-1=0$; Galois group 6T11, of order 48; $h_K=1$; LMFDB 6.2.28037.1; the minimum was proved by Pohst [6]; the next field of signature $(2,2)$ has $|d_K|=29077$

-9747

comment: $x^6-x^5+x^4-2x^3+4x^2-3x+1=0$; Galois group 6T5, of order 18; $h_K=1$; LMFDB 6.0.9747.1; quadratic subfield $\mathbb{Q}(\sqrt{-3})$; the minimum was proved by Liang and Zassenhaus [5], who write the field as $\mathbb{Q}(\theta)$ with $\theta^6-3\theta^5+4\theta^4-4\theta^3+4\theta^2-2\theta+1=0$; the next field of signature $(0,3)$ has $|d_K|=10051$

20134393

comment: $x^7-x^6-6x^5+4x^4+10x^3-4x^2-4x+1=0$; Galois group $S_7$ (7T7); $h_K=1$; LMFDB 7.7.20134393.1; the minimum was proved by Pohst [7]; the next field of signature $(7,0)$ has $|d_K|=25164057$

-2306599

comment: $x^7-3x^5-x^4+x^3+3x^2+x-1=0$; Galois group $S_7$ (7T7); $h_K=1$; LMFDB 7.5.2306599.1; the minimum was proved by Diaz y Diaz [10]; the next field of signature $(5,1)$ has $|d_K|=2369207$

612233

comment: $x^7-x^6+x^5-x^3+x^2-x-1=0$; Galois group $S_7$ (7T7); $h_K=1$; LMFDB 7.3.612233.1; the minimum was proved by Diaz y Diaz [9]; the next field of signature $(3,2)$ has $|d_K|=612569$

-184607

comment: $x^7-x^6-x^5+x^4-x^2+x+1=0$; Galois group $S_7$ (7T7); $h_K=1$; LMFDB 7.1.184607.1; the minimum was proved by Diaz y Diaz [8]; the next field of signature $(1,3)$ has $|d_K|=193327$

282300416

comment: $x^8-4x^7+14x^5-8x^4-12x^3+7x^2+2x-1=0$; Galois group 8T17, of order 32; $h_K=1$; LMFDB 8.8.282300416.1; quadratic subfield $\mathbb{Q}(\sqrt{2})$; $d_K=2^{12}\cdot41^3$; the minimum was proved by Pohst, Martinet and Diaz y Diaz [12]; the next field of signature $(8,0)$ has $|d_K|=309593125$

-65106259

comment: $x^8-5x^6-x^5+7x^4+4x^3-4x^2-2x+1=0$; Galois group $S_8$ (8T50); $h_K=1$; LMFDB 8.6.65106259.1; the minimum was proved by Battistoni [14]; the next field of signature $(6,1)$ has $|d_K|=68494627$

15243125

comment: $x^8-x^7-3x^6+3x^5+3x^4-6x^3-2x^2+3x+1=0$; Galois group 8T17, of order 32; $h_K=1$; LMFDB 8.4.15243125.1; quadratic subfield $\mathbb{Q}(\sqrt{5})$; the minimum was proved by Battistoni [14]; the next field of signature $(4,2)$ has $|d_K|=15297613$

-4286875

comment: $x^8-x^7+x^5-2x^4-x^3+2x^2+2x-1=0$; Galois group the dihedral group of order 16 (8T6); $h_K=1$; LMFDB 8.2.4286875.1; quadratic subfield $\mathbb{Q}(\sqrt{5})$; the minimum was proved by Battistoni [13]; the next field of signature $(2,3)$ has $|d_K|=4296211$

1257728

comment: $x^8-2x^7+4x^5-4x^4+3x^2-2x+1=0$; Galois group 8T17, of order 32; $h_K=1$; LMFDB 8.0.1257728.1; quadratic subfield $\mathbb{Q}(\sqrt{-1})$; $d_K=2^8\cdot17^3$; the minimum was proved by Diaz y Diaz [11]; the next field of signature $(0,4)$ has $|d_K|=1265625$

9685993193

comment: $x^9-9x^7+24x^5-2x^4-20x^3+3x^2+5x-1=0$; Galois group $S_9$ (9T34); $h_K=1$; LMFDB 9.9.9685993193.1; the minimum was proved by Takeuchi [15]; the next field of signature $(9,0)$ has $|d_K|=11779563529$

-109880167

comment: $x^9-2x^8+x^7+x^6-3x^5+x^4+3x^3-2x-1=0$; Galois group $S_9$ (9T34); $h_K=1$; LMFDB 9.3.109880167.1; the minimum was proved by Battistoni [14]; the next field of signature $(3,3)$ has $|d_K|=110852311$

29510281

comment: $x^9-2x^8+2x^7-3x^5+7x^4-8x^3+6x^2-3x+1=0$; Galois group $S_9$ (9T34); $h_K=1$; LMFDB 9.1.29510281.1; the minimum was proved by Battistoni [14]; the next field of signature $(1,4)$ has $|d_K|=30073129$

Definition

Let $K$ be a number field of degree $n$ with $r_1$ real and $r_2$ complex places, $r_1+2r_2=n$, and let $d_K$ be its discriminant [19]. Listed, for each degree $n\geq 2$ and each $r_2$, is the discriminant $d_K$ of the number field $K$ of degree $n$ and signature $(r_1,r_2)$ for which $|d_K|$ is least.

Parameters

$n$

— degree of $K$ ($n\geq 2$)

$r_2$

— number of complex places of $K$ ($0\leq r_2\leq n/2$)

Formulas

(1)

$\operatorname{sign}(d_K)=(-1)^{r_2}$, where $r_2$ is the number of complex places of $K$ [20].

(2)

$|d_K|\geq\left(\frac{n^n}{n!}\right)^2\left(\frac{\pi}{4}\right)^{2r_2}$ for every number field $K$ of degree $n$ with $r_2$ complex places, Minkowski's bound [21]; in particular $|d_K|>1$ for $n\geq 2$.

Comments

(3)

The sign of $d_K$ is $(-1)^{r_2}$ [20], so the value listed is negative exactly when $r_2$ is odd; the minimum is taken over $|d_K|$, and the value is stored with its sign. The absolute values are OEIS A343290 [24], read by rows; their minima over the signatures of one degree are A006557 [25], split into positive and negative discriminants as A343690 [26] and A343772 [27]; and the columns $r_2=0$ and $r_2=1$ are A006554 [28] and A006555 [29].

(4)

Each minimum is attained by a single field up to isomorphism. The comment on each entry gives the reduced defining polynomial $f$ of that field, PARI's polredabs and the polynomial the LMFDB displays, so that $K=\mathbb{Q}(x)$ with $f(x)=0$; the Galois group of the Galois closure, as the transitive group $nTk$ [22]; the class number $h_K$; the LMFDB label; the quadratic subfield if there is one; the paper in which the minimum was proved; and the absolute discriminant of the next field of the same signature. Uniqueness and the next field are read off the LMFDB's list for the signature, which is complete to a bound beyond the minimum in every case here [23].

(5)

In degree $2$ the discriminants are the fundamental discriminants, so the minima are $5$ and $-3$. The two cubic minima are the first entries of the table of regulators of cubic fields, which lists every cubic field with $|D|\leq 3000$. From degree $4$ on each row is a theorem of the geometry of numbers: an element of small trace and small $T_2$-norm generates the field (Hunter, Pohst, Martinet), its minimal polynomial has bounded coefficients, and a finite search closes the signature. Odlyzko's lower bounds [17], from the explicit formula of the Dedekind zeta function, are what keep the searches finite in degrees $7$ to $9$. Voight [16] enumerated every totally real field of root discriminant at most $14$, of which there are $1229$, each of degree at most $9$; the column $r_2=0$ of this table is contained in that list.

(6)

In degree $9$ the signatures $(7,1)$ and $(5,2)$ are open and are not listed. The smallest fields known, in the LMFDB, have $|d_K|=1904081383$ for $(7,1)$ and $|d_K|=453771377$ for $(5,2)$; Odlyzko's unconditional bounds [30] give $|d_K|>1106389454$ and $|d_K|>289155515$ respectively, the best of his Table 4 over its parameter $b$. Battistoni's classification [14] closed the other three signatures of degree $9$ and the last two of degree $8$, and found in signature $(3,3)$ two fields sharing the discriminant $-142989047$ and a field of discriminant $-129079703$ that was not previously known; neither discriminant gives the minimum for signature $(3,3)$. In degree $10$ no signature is settled. $\mathbb{Q}$, of degree $1$ and discriminant $1$, is left out.

(7)

Several of the fields are familiar under other names: $\mathbb{Q}(\sqrt5)$ and $\mathbb{Q}(\sqrt{-3})$ in degree $2$; $\mathbb{Q}(\zeta_7)^+$, of discriminant $49=7^2$, and the field of the plastic number, the real root of $x^3-x-1$, in degree $3$, whose regulators and residues are the first entries of the table of regulators and the table of residues of cubic fields; $\mathbb{Q}(\zeta_{11})^+$, of discriminant $14641=11^4$, in degree $5$; and in degree $6$ the cyclic field of conductor $35$, the compositum of $\mathbb{Q}(\sqrt5)$ and $\mathbb{Q}(\zeta_7)^+$, of discriminant $5^3\cdot7^4$. Every field listed has class number $1$. The three quartic minima have Galois group the dihedral group of order $8$, and the minimum in every signature of degree $7$, and in $(6,1)$, $(1,4)$ and $(3,3)$ of degrees $8$ and $9$, is attained by a field whose Galois group is the full symmetric group.

(8)

Each listed $|d_K|$ exceeds Odlyzko's unconditional lower bound for its signature [30] by a factor between $1.01$ (degree $2$) and $3.4$ (signature $(7,0)$), and exceeds Minkowski's bound (2) by much more; the ratio of the listed $|d_K|$ to the Odlyzko bound grows with $r_1$, which is why the totally real signatures were the last to be settled in each degree.

Programs

(P1)

PARI/GP

nfdisc(x^5 - x^3 - x^2 + x + 1)      \\ 1609, the minimum for signature (1,2)
polsturm(x^5 - x^3 - x^2 + x + 1)    \\ 1 real root, so r_1 = 1 and r_2 = 2

(P2)

Sage

K.<a> = NumberField(x^5 - x^3 - x^2 + x + 1)
K.discriminant(), K.signature()                  # (1609, (1, 2))
from sage.rings.number_field.totallyreal_rel import enumerate_totallyreal_fields_all
enumerate_totallyreal_fields_all(7, 20134393)    # [[20134393, x^7 - x^6 - 6*x^5 + 4*x^4 + 10*x^3 - 4*x^2 - 4*x + 1]]

References

[1]

J. Buchmann and D. Ford, On the computation of totally real quartic fields of small discriminant, Math. Comp. 52 (1989), 161-174 (doi) (zbMATH)

[2]

J. Buchmann, D. Ford and M. Pohst, Enumeration of quartic fields of small discriminant, Math. Comp. 61 (1993), 873-879 (doi) (zbMATH)

[3]

F. Diaz y Diaz, A table of totally real quintic number fields, Math. Comp. 56 (1991), 801-808 (doi) (zbMATH)

[4]

A. Schwarz, M. Pohst and F. Diaz y Diaz, A table of quintic number fields, Math. Comp. 63 (1994), 361-376 (doi) (zbMATH)

[5]

J. Liang and H. Zassenhaus, The minimum discriminant of sixth degree totally complex algebraic number fields, J. Number Theory 9 (1977), 16-35 (doi) (zbMATH)

[6]

M. Pohst, On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory 14 (1982), 99-117 (doi) (zbMATH)

[7]

M. Pohst, The minimum discriminant of seventh degree totally real algebraic number fields, in Number Theory and Algebra (H. Zassenhaus, ed.), Academic Press, 1977, 235-240 (zbMATH)

[8]

F. Diaz y Diaz, Valeurs minima du discriminant des corps de degré 7 ayant une seule place réelle, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 137-139 (zbMATH)

[9]

F. Diaz y Diaz, Valeurs minima du discriminant pour certains types de corps de degré 7, Ann. Inst. Fourier 34 (1984), no. 3, 29-38 (doi) (zbMATH)

[10]

F. Diaz y Diaz, Discriminant minimal et petits discriminants des corps de nombres de degré 7 avec cinq places réelles, J. London Math. Soc. (2) 38 (1988), 33-46 (doi) (zbMATH)

[11]

F. Diaz y Diaz, Petits discriminants des corps de nombres totalement imaginaires de degré 8, J. Number Theory 25 (1987), 34-52 (doi) (zbMATH)

[12]

M. Pohst, J. Martinet and F. Diaz y Diaz, The minimum discriminant of totally real octic fields, J. Number Theory 36 (1990), 145-159 (doi) (zbMATH)

[13]

F. Battistoni, The minimum discriminant of number fields of degree 8 and signature (2,3), J. Number Theory 198 (2019), 386-395 (doi) (zbMATH)

[14]

F. Battistoni, On small discriminants of number fields of degree 8 and 9, J. Théor. Nombres Bordeaux 32 (2020), no. 2, 489-501 (arXiv) (doi) (zbMATH)

[15]

K. Takeuchi, Totally real algebraic number fields of degree 9 with small discriminant, Saitama Math. J. 17 (1999), 63-85 (zbMATH)

[16]

J. Voight, Enumeration of totally real number fields of bounded root discriminant, in Algorithmic Number Theory (ANTS-VIII), Lecture Notes in Computer Science 5011, Springer, 2008, 268-281 (zbMATH)

[17]

A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, Sém. Théor. Nombres Bordeaux (2) 2 (1990), 119-141 (doi) (zbMATH)

[18]

H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer, 1993; Theorem 6.4.2 (Hunter)

Links

[19]

Wikipedia: Discriminant of an algebraic number field

[20]

Wikipedia: Brill's theorem

[21]

Wikipedia: Minkowski's bound

[22]

LMFDB: number fields

[23]

LMFDB: completeness of the number field data, with the bound per degree and signature

[24]

OEIS A343290: minimal absolute discriminant of the number fields of degree $n$ and signature $(n-2k,k)$, read by rows

[25]

OEIS A006557: minimal absolute discriminant of a number field of degree $n$

[26]

OEIS A343690: minimal positive discriminant of a number field of degree $n$

[27]

OEIS A343772: minimal absolute value of a negative discriminant of a number field of degree $n$

[28]

OEIS A006554: minimal discriminant of a totally real number field of degree $n$

[29]

OEIS A006555: minimal absolute discriminant of a number field of degree $n$ with one complex place

[30]

Odlyzko's tables of discriminant bounds (1976); Table 4 holds the unconditional bounds

Similar tables

Regulators of cubic fields — lists every cubic field with $|D|\leq 3000$, so its first entries of each sign are the two cubic rows here

Residues of Dedekind zeta functions of cubic fields — the same cubic fields, indexed by the same discriminant

Residues of Dedekind zeta functions of quadratic fields — indexed by the fundamental discriminants, of which $5$ and $-3$ are the two quadratic rows here

Regulators of real quadratic fields — its first entry is the field of discriminant $5$

Hermite's constants $\gamma_n$ — another record by dimension known in a few cases and bounded elsewhere; Hermite's constant $\gamma_{n-1}$ enters Hunter's bound on the generator of a field of small discriminant

Kissing numbers $\tau_n$ — the corpus's other table of records by dimension with open cases; there the open dimensions carry an interval, here the open signatures are left out

Data properties

Entries are of type: integer

How well the digits are known: exact

Table is complete: no (it holds every signature of degree $2$ to $9$ whose minimum is a theorem, which is all of them but $(7,1)$ and $(5,2)$ in degree $9$, 26 entries in all)

How they were obtained:

Each value is an integer proved to be the minimum in the paper its entry cites, and was read from three independent copies: those papers (Battistoni's Theorem 1 for the four rows of degrees $8$ and $9$ that it settles), OEIS A343290 with the five row and column sequences named in the comments, and the LMFDB's list of fields for each signature, which states the bound to which it is complete.

For every entry the generator recomputes, with PARI, the discriminant of the quoted polynomial (nfdisc), its number of real roots (polsturm), its reduced form (polredabs), the Galois group of its splitting field (polgalois) and its class number (bnfinit certified by bnfcertify), and refuses to return a value on any disagreement. Where a complete enumeration is within reach the minimum is re-derived rather than trusted: for the quadratic rows no fundamental discriminant of the right sign is smaller; for the cubic rows Hunter's box [18] lists every cubic field with smaller $|D|$ and finds none of the signature; for the totally real rows of degree $4$ to $8$ Sage's enumerate_totallyreal_fields_all, an implementation of Voight's algorithm [16] sharing no code with the LMFDB, returns exactly one field up to the value, isomorphic to the quoted one. The mixed signatures from degree $4$ on and all of degree $9$ rest on the cited papers and the LMFDB's completeness statement. Outside the generator, when this was written, every value was also compared with Odlyzko's unconditional lower bound for its signature and with Minkowski's bound, and Liang and Zassenhaus's polynomial for the totally complex sextic field was shown by nfisisom to define the field quoted.