Units and \(S\)-unit groups of number fields

EXAMPLES:

sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^4 - 8*x^2 + 36)
sage: UK = UnitGroup(K); UK
Unit group with structure C4 x Z of
 Number Field in a with defining polynomial x^4 - 8*x^2 + 36

The first generator is a primitive root of unity in the field:

sage: UK.gens()
(u0, u1)
sage: UK.gens_values()  # random
[-1/12*a^3 + 1/6*a, 1/24*a^3 + 1/4*a^2 - 1/12*a - 1]
sage: UK.gen(0).value()
1/12*a^3 - 1/6*a

sage: UK.gen(0)
u0
sage: UK.gen(0) + K.one()   # coerce abstract generator into number field
1/12*a^3 - 1/6*a + 1

sage: [u.multiplicative_order() for u in UK.gens()]
[4, +Infinity]
sage: UK.rank()
1
sage: UK.ngens()
2

Units in the field can be converted into elements of the unit group represented as elements of an abstract multiplicative group:

sage: UK(1)
1
sage: UK(-1)
u0^2
sage: [UK(u) for u in (x^4 - 1).roots(K, multiplicities=False)]
[1, u0^2, u0, u0^3]

sage: UK.fundamental_units() # random
[1/24*a^3 + 1/4*a^2 - 1/12*a - 1]
sage: torsion_gen = UK.torsion_generator(); torsion_gen
u0
sage: torsion_gen.value()
1/12*a^3 - 1/6*a
sage: UK.zeta_order()
4
sage: UK.roots_of_unity()
[1/12*a^3 - 1/6*a, -1, -1/12*a^3 + 1/6*a, 1]

Exp and log functions provide maps between units as field elements and exponent vectors with respect to the generators:

sage: u = UK.exp([13,10]); u # random
-41/8*a^3 - 55/4*a^2 + 41/4*a + 55
sage: UK.log(u)
(1, 10)
sage: u = UK.fundamental_units()[0]
sage: [UK.log(u^k) == (0,k) for k in range(10)]
[True, True, True, True, True, True, True, True, True, True]
sage: all(UK.log(u^k) == (0,k) for k in range(10))
True

sage: K.<a> = NumberField(x^5 - 2,'a')
sage: UK = UnitGroup(K)
sage: UK.rank()
2
sage: UK.fundamental_units()
[a^3 + a^2 - 1, a - 1]

\(S\)-unit groups may be constructed, where \(S\) is a set of primes:

sage: K.<a> = NumberField(x^6 + 2)
sage: S = K.ideal(3).prime_factors(); S
[Fractional ideal (3, a + 1), Fractional ideal (3, a - 1)]
sage: SUK = UnitGroup(K,S=tuple(S)); SUK
S-unit group with structure C2 x Z x Z x Z x Z of
 Number Field in a with defining polynomial x^6 + 2
 with S = (Fractional ideal (3, a + 1), Fractional ideal (3, a - 1))
sage: SUK.primes()
(Fractional ideal (3, a + 1), Fractional ideal (3, a - 1))
sage: SUK.rank()
4
sage: SUK.gens_values()
[-1, a^2 + 1, -a^5 - a^4 + a^2 + a + 1, a + 1, a - 1]
sage: u = 9*prod(SUK.gens_values()); u
-18*a^5 - 18*a^4 - 18*a^3 - 9*a^2 + 9*a + 27
sage: SUK.log(u)
(1, 3, 1, 7, 7)
sage: u == SUK.exp((1,3,1,7,7))
True

A relative number field example:

sage: L.<a, b> = NumberField([x^2 + x + 1, x^4 + 1])
sage: UL = L.unit_group(); UL
Unit group with structure C24 x Z x Z x Z of
 Number Field in a with defining polynomial x^2 + x + 1 over its base field
sage: UL.gens_values() # random
[-b^3*a - b^3, -b^3*a + b, (-b^3 - b^2 - b)*a - b - 1, (-b^3 - 1)*a - b^2 + b - 1]
sage: UL.zeta_order()
24
sage: UL.roots_of_unity()
[-b*a,
 -b^2*a - b^2,
 -b^3,
 -a,
 -b*a - b,
 -b^2,
 b^3*a,
 -a - 1,
 -b,
 b^2*a,
 b^3*a + b^3,
 -1,
 b*a,
 b^2*a + b^2,
 b^3,
 a,
 b*a + b,
 b^2,
 -b^3*a,
 a + 1,
 b,
 -b^2*a,
 -b^3*a - b^3,
 1]

A relative extension example, which worked thanks to the code review by F.W.Clarke:

sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: K.unit_group()
Unit group with structure C2 x Z x Z x Z x Z x Z x Z x Z of Number Field in c
 with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field

AUTHOR:

• John Cremona

class sage.rings.number_field.unit_group.UnitGroup(number_field, proof=True, S=None)

Bases: AbelianGroupWithValues_class

The unit group or an \(S\)-unit group of a number field.

exp(exponents)

Return unit with given exponents with respect to group generators.

INPUT:

• u – Any object from which an element of the unit group’s number field \(K\) may be constructed; an error is raised if an element of \(K\) cannot be constructed from \(u\), or if the element constructed is not a unit.

OUTPUT: a list of integers giving the exponents of \(u\) with respect to the unit group’s basis.

EXAMPLES:

sage: x = polygen(QQ)
sage: K.<z> = CyclotomicField(13)
sage: UK = UnitGroup(K)
sage: [UK.log(u) for u in UK.gens()]
[(1, 0, 0, 0, 0, 0),
 (0, 1, 0, 0, 0, 0),
 (0, 0, 1, 0, 0, 0),
 (0, 0, 0, 1, 0, 0),
 (0, 0, 0, 0, 1, 0),
 (0, 0, 0, 0, 0, 1)]
sage: vec = [65,6,7,8,9,10]
sage: unit = UK.exp(vec)
sage: UK.log(unit)
(13, 6, 7, 8, 9, 10)
sage: u = UK.gens()[-1]
sage: UK.exp(UK.log(u)) == u.value()
True

An S-unit example:

sage: SUK = UnitGroup(K,S=2)
sage: v = (3,1,4,1,5,9,2)
sage: u = SUK.exp(v); u
8732*z^11 - 15496*z^10 - 51840*z^9 - 68804*z^8 - 51840*z^7 - 15496*z^6
 + 8732*z^5 - 34216*z^3 - 64312*z^2 - 64312*z - 34216
sage: SUK.log(u)
(3, 1, 4, 1, 5, 9, 2)
sage: SUK.log(u) == v
True

fundamental_units()

Return generators for the free part of the unit group, as a list.

EXAMPLES:

sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^4 + 23)
sage: U = UnitGroup(K)
sage: U.fundamental_units()  # random
[1/4*a^3 - 7/4*a^2 + 17/4*a - 19/4]

log(u)

Return the exponents of the unit \(u\) with respect to group generators.

INPUT:

• u – Any object from which an element of the unit group’s number field \(K\) may be constructed; an error is raised if an element of \(K\) cannot be constructed from \(u\), or if the element constructed is not a unit.

OUTPUT: a list of integers giving the exponents of \(u\) with respect to the unit group’s basis.

EXAMPLES:

sage: x = polygen(QQ)
sage: K.<z> = CyclotomicField(13)
sage: UK = UnitGroup(K)
sage: [UK.log(u) for u in UK.gens()]
[(1, 0, 0, 0, 0, 0),
 (0, 1, 0, 0, 0, 0),
 (0, 0, 1, 0, 0, 0),
 (0, 0, 0, 1, 0, 0),
 (0, 0, 0, 0, 1, 0),
 (0, 0, 0, 0, 0, 1)]
sage: vec = [65,6,7,8,9,10]
sage: unit = UK.exp(vec); unit  # random
-253576*z^11 + 7003*z^10 - 395532*z^9 - 35275*z^8 - 500326*z^7 - 35275*z^6
 - 395532*z^5 + 7003*z^4 - 253576*z^3 - 59925*z - 59925
sage: UK.log(unit)
(13, 6, 7, 8, 9, 10)

An S-unit example:

sage: SUK = UnitGroup(K, S=2)
sage: v = (3,1,4,1,5,9,2)
sage: u = SUK.exp(v); u
8732*z^11 - 15496*z^10 - 51840*z^9 - 68804*z^8 - 51840*z^7 - 15496*z^6
 + 8732*z^5 - 34216*z^3 - 64312*z^2 - 64312*z - 34216
sage: SUK.log(u)
(3, 1, 4, 1, 5, 9, 2)
sage: SUK.log(u) == v
True

number_field()

Return the number field associated with this unit group.

EXAMPLES:

sage: U = UnitGroup(QuadraticField(-23, 'w')); U
Unit group with structure C2 of
 Number Field in w with defining polynomial x^2 + 23 with w = 4.795831523312720?*I
sage: U.number_field()
Number Field in w with defining polynomial x^2 + 23 with w = 4.795831523312720?*I

primes()

Return the (possibly empty) list of primes associated with this S-unit group.

EXAMPLES:

sage: K.<a> = QuadraticField(-23)
sage: S = tuple(K.ideal(3).prime_factors()); S
(Fractional ideal (3, 1/2*a - 1/2), Fractional ideal (3, 1/2*a + 1/2))
sage: U = UnitGroup(K,S=tuple(S)); U
S-unit group with structure C2 x Z x Z of
 Number Field in a with defining polynomial x^2 + 23 with a = 4.795831523312720?*I
 with S = (Fractional ideal (3, 1/2*a - 1/2), Fractional ideal (3, 1/2*a + 1/2))
sage: U.primes() == S
True

rank()

Return the rank of the unit group.

EXAMPLES:

sage: K.<z> = CyclotomicField(13)
sage: UnitGroup(K).rank()
5
sage: SUK = UnitGroup(K, S=2); SUK.rank()
6

roots_of_unity()

Return all the roots of unity in this unit group, primitive or not.

EXAMPLES:

sage: x = polygen(QQ)
sage: K.<b> = NumberField(x^2 + 1)
sage: U = UnitGroup(K)
sage: zs = U.roots_of_unity(); zs
[b, -1, -b, 1]
sage: [ z**U.zeta_order() for z in zs ]
[1, 1, 1, 1]

torsion_generator()

Return a generator for the torsion part of the unit group.

EXAMPLES:

sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^4 - x^2 + 4)
sage: U = UnitGroup(K)
sage: U.torsion_generator()
u0
sage: U.torsion_generator().value() # random
-1/4*a^3 - 1/4*a + 1/2

zeta(n=2, all=False)

Return one, or a list of all, primitive \(n\)-th root of unity in this unit group.

EXAMPLES:

sage: x = polygen(QQ)
sage: K.<z> = NumberField(x^2 + 3)
sage: U = UnitGroup(K)
sage: U.zeta(1)
1
sage: U.zeta(2)
-1
sage: U.zeta(2, all=True)
[-1]
sage: U.zeta(3)
-1/2*z - 1/2
sage: U.zeta(3, all=True)
[-1/2*z - 1/2, 1/2*z - 1/2]
sage: U.zeta(4)
Traceback (most recent call last):
...
ValueError: n (=4) does not divide order of generator

sage: r.<x> = QQ[]
sage: K.<b> = NumberField(x^2 + 1)
sage: U = UnitGroup(K)
sage: U.zeta(4)
b
sage: U.zeta(4,all=True)
[b, -b]
sage: U.zeta(3)
Traceback (most recent call last):
...
ValueError: n (=3) does not divide order of generator
sage: U.zeta(3, all=True)
[]

zeta_order()

Returns the order of the torsion part of the unit group.

EXAMPLES:

sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^4 - x^2 + 4)
sage: U = UnitGroup(K)
sage: U.zeta_order()
6