Fields

class sage.categories.fields.Fields(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton

The category of (commutative) fields, i.e. commutative rings where all non-zero elements have multiplicative inverses

EXAMPLES:

sage: K = Fields()
sage: K
Category of fields
sage: Fields().super_categories()
[Category of euclidean domains, Category of division rings]

sage: K(IntegerRing())
Rational Field
sage: K(PolynomialRing(GF(3), 'x'))
Fraction Field of Univariate Polynomial Ring in x over
Finite Field of size 3
sage: K(RealField())
Real Field with 53 bits of precision
class ElementMethods
euclidean_degree()

Return the degree of this element as an element of an Euclidean domain.

In a field, this returns 0 for all but the zero element (for which it is undefined).

EXAMPLES:

sage: QQ.one().euclidean_degree()
0
factor()

Return a factorization of self.

Since self is either a unit or zero, this function is trivial.

EXAMPLES:

sage: x = GF(7)(5)
sage: x.factor()
5
sage: RR(0).factor()
Traceback (most recent call last):
...
ArithmeticError: factorization of 0.000000000000000 is not defined
gcd(other)

Greatest common divisor.

Note

Since we are in a field and the greatest common divisor is only determined up to a unit, it is correct to either return zero or one. Note that fraction fields of unique factorization domains provide a more sophisticated gcd.

EXAMPLES:

sage: K = GF(5)
sage: K(2).gcd(K(1))
1
sage: K(0).gcd(K(0))
0
sage: all(x.gcd(y) == (0 if x == 0 and y == 0 else 1) for x in K for y in K)
True

For field of characteristic zero, the gcd of integers is considered as if they were elements of the integer ring:

sage: gcd(15.0,12.0)
3.00000000000000

But for others floating point numbers, the gcd is just \(0.0\) or \(1.0\):

sage: gcd(3.2, 2.18)
1.00000000000000

sage: gcd(0.0, 0.0)
0.000000000000000

AUTHOR:

is_unit()

Returns True if self has a multiplicative inverse.

EXAMPLES:

sage: QQ(2).is_unit()
True
sage: QQ(0).is_unit()
False
lcm(other)

Least common multiple.

Note

Since we are in a field and the least common multiple is only determined up to a unit, it is correct to either return zero or one. Note that fraction fields of unique factorization domains provide a more sophisticated lcm.

EXAMPLES:

sage: GF(2)(1).lcm(GF(2)(0))
0
sage: GF(2)(1).lcm(GF(2)(1))
1

For field of characteristic zero, the lcm of integers is considered as if they were elements of the integer ring:

sage: lcm(15.0,12.0)
60.0000000000000

But for others floating point numbers, it is just \(0.0\) or \(1.0\):

sage: lcm(3.2, 2.18)
1.00000000000000

sage: lcm(0.0, 0.0)
0.000000000000000

AUTHOR:

quo_rem(other)

Return the quotient with remainder of the division of this element by other.

INPUT:

  • other – an element of the field

EXAMPLES:

sage: f,g = QQ(1), QQ(2)
sage: f.quo_rem(g)
(1/2, 0)
xgcd(other)

Compute the extended gcd of self and other.

INPUT:

  • other – an element with the same parent as self

OUTPUT:

A tuple (r, s, t) of elements in the parent of self such that r = s * self + t * other. Since the computations are done over a field, r is zero if self and other are zero, and one otherwise.

AUTHORS:

EXAMPLES:

sage: K = GF(5)
sage: K(2).xgcd(K(1))
(1, 3, 0)
sage: K(0).xgcd(K(4))
(1, 0, 4)
sage: K(1).xgcd(K(1))
(1, 1, 0)
sage: GF(5)(0).xgcd(GF(5)(0))
(0, 0, 0)

The xgcd of non-zero floating point numbers will be a triple of floating points. But if the input are two integral floating points the result is a floating point version of the standard gcd on \(\ZZ\):

sage: xgcd(12.0, 8.0)
(4.00000000000000, 1.00000000000000, -1.00000000000000)

sage: xgcd(3.1, 2.98714)
(1.00000000000000, 0.322580645161290, 0.000000000000000)

sage: xgcd(0.0, 1.1)
(1.00000000000000, 0.000000000000000, 0.909090909090909)
Fields.Finite

alias of FiniteFields

class Fields.ParentMethods
fraction_field()

Returns the fraction field of self, which is self.

EXAMPLES:

sage: QQ.fraction_field() is QQ
True
is_field(proof=True)

Returns True as self is a field.

EXAMPLES:

sage: QQ.is_field()
True
sage: Parent(QQ,category=Fields()).is_field()
True
is_integrally_closed()

Return True, as per IntegralDomain.is_integrally_closed(): for every field \(F\), \(F\) is its own field of fractions, hence every element of \(F\) is integral over \(F\).

EXAMPLES:

sage: QQ.is_integrally_closed()
True
sage: QQbar.is_integrally_closed()
True
sage: Z5 = GF(5); Z5
Finite Field of size 5
sage: Z5.is_integrally_closed()
True
is_perfect()

Return whether this field is perfect, i.e., its characteristic is \(p=0\) or every element has a \(p\)-th root.

EXAMPLES:

sage: QQ.is_perfect()
True
sage: GF(2).is_perfect()
True
sage: FunctionField(GF(2), 'x').is_perfect()
False
Fields.extra_super_categories()

EXAMPLES:

sage: Fields().extra_super_categories()
[Category of euclidean domains]