Module number_field

Return the unique copy of the gp (PARI) interpreter
used for number field computations.

EXAMPLES:
    sage: from sage.rings.number_field.number_field import gp
    sage: gp()
    GP/PARI interpreter

Return \emph{the} number field defined by the given irreducible
polynomial and with variable with the given name.  If check is
True (the default), also verify that the defining polynomial is
irreducible and over Q.

INPUT:
    polynomial -- a polynomial over QQ or a number field, or
                  a list of polynomials.
    name -- a string (default: 'a'), the name of the generator
    check -- bool (default: True); do type checking and
             irreducibility checking.

EXAMPLES:
    sage: z = QQ['z'].0
    sage: K = NumberField(z^2 - 2,'s'); K
    Number Field in s with defining polynomial z^2 - 2
    sage: s = K.0; s
    s
    sage: s*s    
    2
    sage: s^2
    2

EXAMPLES: Constructing a relative number field
    sage: K.<a> = NumberField(x^2 - 2)
    sage: R.<t> = K[]
    sage: L.<b> = K.extension(t^3+t+a); L
    Number Field in b with defining polynomial t^3 + t + a over its base field
    sage: L.absolute_field('c')
    Number Field in c with defining polynomial x^6 + 2*x^4 + x^2 - 2
    sage: a*b
    a*b
    sage: L(a)
    a
    sage: L.lift_to_base(b^3 + b)
    -a

Constructing another number field:
    sage: k.<i> = NumberField(x^2 + 1)
    sage: R.<z> = k[]
    sage: m.<j> = NumberField(z^3 + i*z + 3)
    sage: m
    Number Field in j with defining polynomial z^3 + i*z + 3 over its base field

Number fields are globally unique.
    sage: K.<a>= NumberField(x^3-5)     
    sage: a^3
    5
    sage: L.<a>= NumberField(x^3-5)
    sage: K is L
    True

Having different defining polynomials makes them fields different:
    sage: x = polygen(QQ, 'x'); y = polygen(QQ, 'y')
    sage: k.<a> = NumberField(x^2 + 3)
    sage: m.<a> = NumberField(y^2 + 3)
    sage: k
    Number Field in a with defining polynomial x^2 + 3
    sage: m
    Number Field in a with defining polynomial y^2 + 3

An example involving a variable name that defines a function in
PARI:
    sage: theta = polygen(QQ, 'theta')
    sage: M.<z> = NumberField([theta^3 + 4, theta^2 + 3]); M
    Number Field in z0 with defining polynomial theta^3 + 4 over its base field        

Return the tower of number fields defined by the polynomials or
number fields in the list v.

This is the field constructed first from v[0], then over that
field from v[1], etc.  If all is False, then each v[i] must be
irreducible over the previous fields.  Otherwise a list of all
possible fields defined by all those polynomials is output.

If names defines a variable name a, say, then the generators of
the intermediate number fields are a0, a1, a2, ...

INPUT:
    v -- a list of polynomials or number fields
    names -- variable names
    check -- bool (default: True) only relevant if all is False.
           Then check irreducibility of each input polynomial.
           
OUTPUT:
    a single number field or a list of number fields

EXAMPLES:
    sage: k.<a,b,c> = NumberField([x^2 + 1, x^2 + 3, x^2 + 5]); k
    Number Field in a with defining polynomial x^2 + 1 over its base field
    sage: a^2
    -1
    sage: b^2
    -3
    sage: c^2
    -5
    sage: (a+b+c)^2
    (2*b + 2*c)*a + 2*c*b - 9

The Galois group is a product of 3 groups of order 2:
    sage: k.galois_group()
    Galois group PARI group [8, 1, 3, "E(8)=2[x]2[x]2"] of degree 8 of the Number Field in a with defining polynomial x^2 + 1 over its base field
    

Repeatedly calling base_field allows us to descend the internally
constructed tower of fields:
    sage: k.base_field()
    Number Field in b with defining polynomial x^2 + 3 over its base field
    sage: k.base_field().base_field()
    Number Field in c with defining polynomial x^2 + 5
    sage: k.base_field().base_field().base_field()
    Rational Field

In the following example the second polynomial is reducible over the
first, so we get an error:
    sage: v = NumberField([x^3 - 2, x^3 - 2], names='a')
    Traceback (most recent call last):
    ...
    ValueError: defining polynomial (x^3 - 2) must be irreducible

We mix polynomial parent rings:
    sage: k.<y> = QQ[]
    sage: m = NumberField([y^3 - 3, x^2 + x + 1, y^3 + 2], 'beta')
    sage: m
    Number Field in beta0 with defining polynomial y^3 - 3 over its base field
    sage: m.base_field ()
    Number Field in beta1 with defining polynomial x^2 + x + 1 over its base field

A tower of quadratic fields:
    sage: K.<a> = NumberField([x^2 + 3, x^2 + 2, x^2 + 1])
    sage: K
    Number Field in a0 with defining polynomial x^2 + 3 over its base field
    sage: K.base_field()
    Number Field in a1 with defining polynomial x^2 + 2 over its base field
    sage: K.base_field().base_field()
    Number Field in a2 with defining polynomial x^2 + 1

A bigger tower of quadratic fields.
    sage: K.<a2,a3,a5,a7> = NumberField([x^2 + p for p in [2,3,5,7]]); K
    Number Field in a2 with defining polynomial x^2 + 2 over its base field
    sage: a2^2
    -2
    sage: a3^2
    -3
    sage: (a2+a3+a5+a7)^3
    ((6*a5 + 6*a7)*a3 + 6*a7*a5 - 47)*a2 + (6*a7*a5 - 45)*a3 + (-41)*a5 - 37*a7

Return a quadratic field obtained by adjoining a square root of
$D$ to the rational numbers, where $D$ is not a perfect square.

INPUT:
    D -- a rational number
    name -- variable name
    check -- bool (default: True)

OUTPUT:
    A number field defined by a quadratic polynomial.

EXAMPLES:
    sage: QuadraticField(3, 'a')
    Number Field in a with defining polynomial x^2 - 3
    sage: K.<theta> = QuadraticField(3); K
    Number Field in theta with defining polynomial x^2 - 3
    sage: QuadraticField(9, 'a')
    Traceback (most recent call last):
    ...
    ValueError: D must not be a perfect square.
    sage: QuadraticField(9, 'a', check=False)
    Number Field in a with defining polynomial x^2 - 9

Quadratic number fields derive from general number fields.
    sage: type(K)
    <class 'sage.rings.number_field.number_field.NumberField_quadratic'>
    sage: is_NumberField(K)
    True

Return True if x is an absolute number field. 

EXAMPLES:
    sage: is_AbsoluteNumberField(NumberField(x^2+1,'a'))
    True
    sage: is_AbsoluteNumberField(NumberField([x^3 + 17, x^2+1],'a'))
    False

The rationals are a number field, but they're not of the absolute number field class. 
    sage: is_AbsoluteNumberField(QQ)
    False

Return True if x is of the quadratic \emph{number} field type.

EXAMPLES:
    sage: is_QuadraticField(QuadraticField(5,'a'))
    True
    sage: is_QuadraticField(NumberField(x^2 - 5, 'b'))
    True
    sage: is_QuadraticField(NumberField(x^3 - 5, 'b'))
    False

A quadratic field specially refers to a number field, not a finite
field:
    sage: is_QuadraticField(GF(9,'a'))
    False

Return True if x is a relative number field.

EXAMPLES:
    sage: is_RelativeNumberField(NumberField(x^2+1,'a'))
    False
    sage: k.<a> = NumberField(x^3 - 2)
    sage: l.<b> = k.extension(x^3 - 3); l
    Number Field in b with defining polynomial x^3 - 3 over its base field
    sage: is_RelativeNumberField(l)
    True
    sage: is_RelativeNumberField(QQ)
    False        

Return the n-th cyclotomic field, where n is a positive integer.

INPUT:
    n -- a positive integer
    names -- name of generator (optional -- defaults to zetan).

EXAMPLES:
We create the $7$th cyclotomic field $\QQ(\zeta_7)$ with the
default generator name.
    sage: k = CyclotomicField(7); k
    Cyclotomic Field of order 7 and degree 6
    sage: k.gen()
    zeta7

Cyclotomic fields are of a special type.
    sage: type(k)
    <class 'sage.rings.number_field.number_field.NumberField_cyclotomic'>

We can specify a different generator name as follows.
    sage: k.<z7> = CyclotomicField(7); k
    Cyclotomic Field of order 7 and degree 6
    sage: k.gen()
    z7

The $n$ must be an integer. 
    sage: CyclotomicField(3/2)
    Traceback (most recent call last):
    ...
    TypeError: no coercion of this rational to integer

The degree must be positive. 
    sage: CyclotomicField(0)
    Traceback (most recent call last):
    ...
    ValueError: n (=0) must be a positive integer

The special case $n=1$ does \emph{not} return the rational numbers:
    sage: CyclotomicField(1)
    Cyclotomic Field of order 1 and degree 1

    sage: cf6 = CyclotomicField(6) ; z6 = cf6.0
    sage: cf3 = CyclotomicField(3) ; z3 = cf3.0
    sage: cf3(z6)
    zeta3 + 1
    sage: cf6(z3)
    zeta6 - 1
    sage: cf9 = CyclotomicField(9) ; z9 = cf9.0
    sage: cf18 = CyclotomicField(18) ; z18 = cf18.0
    sage: cf18(z9)
    zeta18^2
    sage: cf9(z18)
    -zeta9^5
    sage: cf18(z3)
    zeta18^3 - 1
    sage: cf18(z6)
    zeta18^3
    sage: cf18(z6)**2
    zeta18^3 - 1
    sage: cf9(z3)
    zeta9^3

Return True if x is a cyclotomic field, i.e., of the special
cyclotomic field class.  This function does not return True
for a number field that just happens to be isomorphic to a
cyclotomic field. 

EXAMPLES:
    sage: is_CyclotomicField(NumberField(x^2 + 1,'zeta4'))
    False
    sage: is_CyclotomicField(CyclotomicField(4))
    True
    sage: is_CyclotomicField(CyclotomicField(1))
    True
    sage: is_CyclotomicField(QQ)
    False
    sage: is_CyclotomicField(7)
    False

Return True if the integer $D$ is a fundamental discriminant, i.e.,
if $D \con 0,1\pmod{4}$, and $D\neq 0, 1$ and either (1) $D$ is square free
or (2) we have $D\con 0\pmod{4}$ with $D/4 \con 2,3\pmod{4}$ and $D/4$
square free.  These are exactly the discriminants of quadratic fields.

EXAMPLES:
    sage: [D for D in range(-15,15) if is_fundamental_discriminant(D)]
    [-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]
    sage: [D for D in range(-15,15) if not is_square(D) and QuadraticField(D,'a').disc() == D]
    [-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]

This is used in pickling generic number fields.

EXAMPLES:
    sage: from sage.rings.number_field.number_field import NumberField_generic_v1
    sage: R.<x> = QQ[]
    sage: NumberField_generic_v1(x^2 + 1, 'i', 'i')
    Number Field in i with defining polynomial x^2 + 1

This is used in pickling generic number fields.

EXAMPLES:
    sage: from sage.rings.number_field.number_field import NumberField_generic_v1
    sage: R.<x> = QQ[]
    sage: NumberField_generic_v1(x^2 + 1, 'i', 'i')
    Number Field in i with defining polynomial x^2 + 1

This is used in pickling relative fields.

EXAMPLES:
    sage: from sage.rings.number_field.number_field import NumberField_relative_v1
    sage: R.<x> = CyclotomicField(3)[]
    sage: NumberField_relative_v1(CyclotomicField(3), x^2 + 7, 'a', 'a')
    Number Field in a with defining polynomial x^2 + 7 over its base field

This is used in pickling relative fields.

EXAMPLES:
    sage: from sage.rings.number_field.number_field import NumberField_relative_v1
    sage: R.<x> = CyclotomicField(3)[]
    sage: NumberField_relative_v1(CyclotomicField(3), x^2 + 7, 'a', 'a')
    Number Field in a with defining polynomial x^2 + 7 over its base field

This is used in pickling cyclotomic fields. 

EXAMPLES:
    sage: from sage.rings.number_field.number_field import NumberField_cyclotomic_v1
    sage: NumberField_cyclotomic_v1(5,'a')
    Cyclotomic Field of order 5 and degree 4
    sage: NumberField_cyclotomic_v1(5,'a').variable_name()
    'a'

This is used in pickling quadratic fields. 

EXAMPLES:
    sage: from sage.rings.number_field.number_field import NumberField_quadratic_v1
    sage: R.<x> = QQ[]
    sage: NumberField_quadratic_v1(x^2 - 2, 'd')
    Number Field in d with defining polynomial x^2 - 2

ZZ

Value:

['4ti2-20061025', 'R-2.6.0', 'atlas-3.7.37', 'atlas-3.8.1', 'atlas-3.8.1.p1', 'atlas-3.8.1.p3', 'atlas-3.8.p11', 'atlas-3.8.p6', ...