Base class for finite field elements¶

AUTHORS:

David Roe (2010-01-14): factored out of sage.structure.element
Sebastian Oehms (2018-07-19): added conjugate() (see Issue #26761)

class sage.rings.finite_rings.element_base.Cache_base[source]¶

Bases: SageObject

fetch_int(number)[source]¶

Given an integer less than \(p^n\) with base \(2\) representation \(a_0 + a_1 \cdot 2 + \cdots + a_k 2^k\), this returns \(a_0 + a_1 x + \cdots + a_k x^k\), where \(x\) is the generator of this finite field.

EXAMPLES:

Sage

sage: k.<a> = GF(2^48)
sage: k._cache.fetch_int(2^33 + 2 + 1)                                      # needs sage.libs.ntl
a^33 + a + 1

Python

>>> from sage.all import *
>>> k = GF(Integer(2)**Integer(48), names=('a',)); (a,) = k._first_ngens(1)
>>> k._cache.fetch_int(Integer(2)**Integer(33) + Integer(2) + Integer(1))                                      # needs sage.libs.ntl
a^33 + a + 1

class sage.rings.finite_rings.element_base.FinitePolyExtElement[source]¶

Bases: FiniteRingElement

Elements represented as polynomials modulo a given ideal.

additive_order()[source]¶

Return the additive order of this finite field element.

EXAMPLES:

Sage

sage: k.<a> = FiniteField(2^12, 'a')
sage: b = a^3 + a + 1
sage: b.additive_order()
2
sage: k(0).additive_order()
1

Python

>>> from sage.all import *
>>> k = FiniteField(Integer(2)**Integer(12), 'a', names=('a',)); (a,) = k._first_ngens(1)
>>> b = a**Integer(3) + a + Integer(1)
>>> b.additive_order()
2
>>> k(Integer(0)).additive_order()
1

charpoly(var='x', algorithm='pari')[source]¶

Return the characteristic polynomial of self as a polynomial with given variable.

INPUT:

var – string (default: 'x')
algorithm – string (default: 'pari'):
- 'pari' – use pari’s charpoly
- 'matrix' – return the charpoly computed from the matrix of left multiplication by self

The result is not cached.

EXAMPLES:

Sage

sage: from sage.rings.finite_rings.element_base import FinitePolyExtElement
sage: k.<a> = FiniteField(19^2)
sage: parent(a)
Finite Field in a of size 19^2
sage: b = a**20
sage: p = FinitePolyExtElement.charpoly(b, "x", algorithm='pari')
sage: q = FinitePolyExtElement.charpoly(b, "x", algorithm='matrix')         # needs sage.modules
sage: q == p                                                                # needs sage.modules
True
sage: p
x^2 + 15*x + 4
sage: factor(p)
(x + 17)^2
sage: b.minpoly('x')
x + 17

Python

>>> from sage.all import *
>>> from sage.rings.finite_rings.element_base import FinitePolyExtElement
>>> k = FiniteField(Integer(19)**Integer(2), names=('a',)); (a,) = k._first_ngens(1)
>>> parent(a)
Finite Field in a of size 19^2
>>> b = a**Integer(20)
>>> p = FinitePolyExtElement.charpoly(b, "x", algorithm='pari')
>>> q = FinitePolyExtElement.charpoly(b, "x", algorithm='matrix')         # needs sage.modules
>>> q == p                                                                # needs sage.modules
True
>>> p
x^2 + 15*x + 4
>>> factor(p)
(x + 17)^2
>>> b.minpoly('x')
x + 17

conjugate()[source]¶

This methods returns the result of the Frobenius morphism in the case where the field is a quadratic extension, say \(GF(q^2)\), where \(q=p^k\) is a prime power and \(p\) the characteristic of the field.

OUTPUT:

Instance of this class representing the image under the Frobenius morphisms.

EXAMPLES:

Sage

sage: F.<a> = GF(16)
sage: b = a.conjugate(); b
a + 1
sage: a == b.conjugate()
True

sage: F.<a> = GF(27)
sage: a.conjugate()
Traceback (most recent call last):
...
TypeError: cardinality of the field must be a square number

Python

>>> from sage.all import *
>>> F = GF(Integer(16), names=('a',)); (a,) = F._first_ngens(1)
>>> b = a.conjugate(); b
a + 1
>>> a == b.conjugate()
True

>>> F = GF(Integer(27), names=('a',)); (a,) = F._first_ngens(1)
>>> a.conjugate()
Traceback (most recent call last):
...
TypeError: cardinality of the field must be a square number

frobenius(k=1)[source]¶

Return the \((p^k)\)-th power of self, where \(p\) is the characteristic of the field.

INPUT:

k – integer (default: 1, must fit in C int type)

Note that if \(k\) is negative, then this computes the appropriate root.

EXAMPLES:

Sage

sage: F.<a> = GF(29^2)
sage: z = a^2 + 5*a + 1
sage: z.pth_power()
19*a + 20
sage: z.pth_power(10)
10*a + 28
sage: z.pth_power(-10) == z
True
sage: F.<b> = GF(2^12)
sage: y = b^3 + b + 1
sage: y == (y.pth_power(-3))^(2^3)
True
sage: y.pth_power(2)
b^7 + b^6 + b^5 + b^4 + b^3 + b

Python

>>> from sage.all import *
>>> F = GF(Integer(29)**Integer(2), names=('a',)); (a,) = F._first_ngens(1)
>>> z = a**Integer(2) + Integer(5)*a + Integer(1)
>>> z.pth_power()
19*a + 20
>>> z.pth_power(Integer(10))
10*a + 28
>>> z.pth_power(-Integer(10)) == z
True
>>> F = GF(Integer(2)**Integer(12), names=('b',)); (b,) = F._first_ngens(1)
>>> y = b**Integer(3) + b + Integer(1)
>>> y == (y.pth_power(-Integer(3)))**(Integer(2)**Integer(3))
True
>>> y.pth_power(Integer(2))
b^7 + b^6 + b^5 + b^4 + b^3 + b

is_square()[source]¶

Return True if and only if this element is a perfect square.

EXAMPLES:

Sage

sage: k.<a> = FiniteField(9, impl='givaro', modulus='primitive')            # needs sage.libs.linbox
sage: a.is_square()                                                         # needs sage.libs.linbox
False
sage: (a**2).is_square()                                                    # needs sage.libs.linbox
True
sage: k.<a> = FiniteField(4, impl='ntl', modulus='primitive')               # needs sage.libs.ntl
sage: (a**2).is_square()                                                    # needs sage.libs.ntl
True
sage: k.<a> = FiniteField(17^5, impl='pari_ffelt', modulus='primitive')     # needs sage.libs.pari
sage: a.is_square()                                                         # needs sage.libs.pari
False
sage: (a**2).is_square()                                                    # needs sage.libs.pari
True

Python

>>> from sage.all import *
>>> k = FiniteField(Integer(9), impl='givaro', modulus='primitive', names=('a',)); (a,) = k._first_ngens(1)# needs sage.libs.linbox
>>> a.is_square()                                                         # needs sage.libs.linbox
False
>>> (a**Integer(2)).is_square()                                                    # needs sage.libs.linbox
True
>>> k = FiniteField(Integer(4), impl='ntl', modulus='primitive', names=('a',)); (a,) = k._first_ngens(1)# needs sage.libs.ntl
>>> (a**Integer(2)).is_square()                                                    # needs sage.libs.ntl
True
>>> k = FiniteField(Integer(17)**Integer(5), impl='pari_ffelt', modulus='primitive', names=('a',)); (a,) = k._first_ngens(1)# needs sage.libs.pari
>>> a.is_square()                                                         # needs sage.libs.pari
False
>>> (a**Integer(2)).is_square()                                                    # needs sage.libs.pari
True

Sage

sage: k(0).is_square()                                                      # needs sage.libs.linbox
True

Python

>>> from sage.all import *
>>> k(Integer(0)).is_square()                                                      # needs sage.libs.linbox
True

list()[source]¶

Return the list of coefficients (in little-endian) of this finite field element when written as a polynomial in the generator.

Equivalent to calling list() on this element.

EXAMPLES:

Sage

sage: x = polygen(GF(71))
sage: F.<u> = GF(71^7, modulus=x^7 + x + 1)
sage: a = 3 + u + 3*u^2 + 3*u^3 + 7*u^4
sage: a.list()
[3, 1, 3, 3, 7, 0, 0]
sage: a.list() == list(a) == [a[i] for i in range(F.degree())]
True

Python

>>> from sage.all import *
>>> x = polygen(GF(Integer(71)))
>>> F = GF(Integer(71)**Integer(7), modulus=x**Integer(7) + x + Integer(1), names=('u',)); (u,) = F._first_ngens(1)
>>> a = Integer(3) + u + Integer(3)*u**Integer(2) + Integer(3)*u**Integer(3) + Integer(7)*u**Integer(4)
>>> a.list()
[3, 1, 3, 3, 7, 0, 0]
>>> a.list() == list(a) == [a[i] for i in range(F.degree())]
True

The coefficients returned are those of a fully reduced representative of the finite field element:

Sage

sage: b = u^777
sage: b.list()
[9, 69, 4, 27, 40, 10, 56]
sage: (u.polynomial()^777).list()
[0, 0, 0, 0, ..., 0, 1]

Python

>>> from sage.all import *
>>> b = u**Integer(777)
>>> b.list()
[9, 69, 4, 27, 40, 10, 56]
>>> (u.polynomial()**Integer(777)).list()
[0, 0, 0, 0, ..., 0, 1]

matrix(reverse=False)[source]¶

Return the matrix of left multiplication by the element on the power basis \(1, x, x^2, \ldots, x^{d-1}\) for the field extension.

Thus the emph{columns} of this matrix give the images of each of the \(x^i\).

INPUT:

reverse – if True, act on vectors in reversed order

EXAMPLES:

Sage

sage: # needs sage.modules
sage: k.<a> = GF(2^4)
sage: b = k.random_element()
sage: vector(a*b) == a.matrix() * vector(b)
True
sage: (a*b)._vector_(reverse=True) == a.matrix(reverse=True) * b._vector_(reverse=True)
True

Python

>>> from sage.all import *
>>> # needs sage.modules
>>> k = GF(Integer(2)**Integer(4), names=('a',)); (a,) = k._first_ngens(1)
>>> b = k.random_element()
>>> vector(a*b) == a.matrix() * vector(b)
True
>>> (a*b)._vector_(reverse=True) == a.matrix(reverse=True) * b._vector_(reverse=True)
True

minimal_polynomial(var='x')[source]¶

Return the minimal polynomial of this element (over the corresponding prime subfield).

EXAMPLES:

Sage

sage: k.<a> = FiniteField(3^4)
sage: parent(a)
Finite Field in a of size 3^4
sage: b=a**20;p=charpoly(b,"y");p
y^4 + 2*y^2 + 1
sage: factor(p)
(y^2 + 1)^2
sage: b.minimal_polynomial('y')
y^2 + 1

Python

>>> from sage.all import *
>>> k = FiniteField(Integer(3)**Integer(4), names=('a',)); (a,) = k._first_ngens(1)
>>> parent(a)
Finite Field in a of size 3^4
>>> b=a**Integer(20);p=charpoly(b,"y");p
y^4 + 2*y^2 + 1
>>> factor(p)
(y^2 + 1)^2
>>> b.minimal_polynomial('y')
y^2 + 1

minpoly(var='x', algorithm='pari')[source]¶

Return the minimal polynomial of this element (over the corresponding prime subfield).

INPUT:

var – string (default: 'x')
algorithm – string (default: 'pari'):
- 'pari' – use pari’s minpoly
- 'matrix' – return the minpoly computed from the matrix of left multiplication by self

EXAMPLES:

Sage

sage: from sage.rings.finite_rings.element_base import FinitePolyExtElement
sage: k.<a> = FiniteField(19^2)
sage: parent(a)
Finite Field in a of size 19^2
sage: b=a**20
sage: p=FinitePolyExtElement.minpoly(b,"x", algorithm='pari')
sage: q=FinitePolyExtElement.minpoly(b,"x", algorithm='matrix')
sage: q == p
True
sage: p
x + 17

Python

>>> from sage.all import *
>>> from sage.rings.finite_rings.element_base import FinitePolyExtElement
>>> k = FiniteField(Integer(19)**Integer(2), names=('a',)); (a,) = k._first_ngens(1)
>>> parent(a)
Finite Field in a of size 19^2
>>> b=a**Integer(20)
>>> p=FinitePolyExtElement.minpoly(b,"x", algorithm='pari')
>>> q=FinitePolyExtElement.minpoly(b,"x", algorithm='matrix')
>>> q == p
True
>>> p
x + 17

multiplicative_order()[source]¶

Return the multiplicative order of this field element.

EXAMPLES:

Sage

sage: S.<a> = GF(5^3); S
Finite Field in a of size 5^3
sage: a.multiplicative_order()
124
sage: (a^8).multiplicative_order()
31
sage: S(0).multiplicative_order()
Traceback (most recent call last):
...
ArithmeticError: Multiplicative order of 0 not defined.

Python

>>> from sage.all import *
>>> S = GF(Integer(5)**Integer(3), names=('a',)); (a,) = S._first_ngens(1); S
Finite Field in a of size 5^3
>>> a.multiplicative_order()
124
>>> (a**Integer(8)).multiplicative_order()
31
>>> S(Integer(0)).multiplicative_order()
Traceback (most recent call last):
...
ArithmeticError: Multiplicative order of 0 not defined.

norm()[source]¶

Return the norm of self down to the prime subfield.

This is the product of the Galois conjugates of self.

EXAMPLES:

Sage

sage: S.<b> = GF(5^2); S
Finite Field in b of size 5^2
sage: b.norm()
2
sage: b.charpoly('t')
t^2 + 4*t + 2

Python

>>> from sage.all import *
>>> S = GF(Integer(5)**Integer(2), names=('b',)); (b,) = S._first_ngens(1); S
Finite Field in b of size 5^2
>>> b.norm()
2
>>> b.charpoly('t')
t^2 + 4*t + 2

Next we consider a cubic extension:

Sage

sage: S.<a> = GF(5^3); S
Finite Field in a of size 5^3
sage: a.norm()
2
sage: a.charpoly('t')
t^3 + 3*t + 3
sage: a * a^5 * (a^25)
2

Python

>>> from sage.all import *
>>> S = GF(Integer(5)**Integer(3), names=('a',)); (a,) = S._first_ngens(1); S
Finite Field in a of size 5^3
>>> a.norm()
2
>>> a.charpoly('t')
t^3 + 3*t + 3
>>> a * a**Integer(5) * (a**Integer(25))
2

nth_root(n, extend=False, all=False, algorithm=None, cunningham=False)[source]¶

Return an \(n\)-th root of self.

INPUT:

n – integer \(\geq 1\)
extend – boolean (default: False); if True, return an \(n\)-th root in an extension ring, if necessary. Otherwise, raise a ValueError if the root is not in the base ring. Warning: this option is not implemented!
all – boolean (default: False); if True, return all \(n\)-th roots of self, instead of just one
algorithm – string (default: None); 'Johnston' is the only currently supported option. For IntegerMod elements, the problem is reduced to the prime modulus case using CRT and \(p\)-adic logs, and then this algorithm used.

OUTPUT:

If self has an \(n\)-th root, returns one (if all is False) or a list of all of them (if all is True). Otherwise, raises a ValueError (if extend is False) or a NotImplementedError (if extend is True).

Warning

The extend option is not implemented (yet).

EXAMPLES:

Sage

sage: K = GF(31)
sage: a = K(22)
sage: K(22).nth_root(7)
13
sage: K(25).nth_root(5)
5
sage: K(23).nth_root(3)
29

sage: K.<a> = GF(625)
sage: (3*a^2+a+1).nth_root(13)**13
3*a^2 + a + 1

sage: k.<a> = GF(29^2)
sage: b = a^2 + 5*a + 1
sage: b.nth_root(11)
3*a + 20
sage: b.nth_root(5)
Traceback (most recent call last):
...
ValueError: no nth root
sage: b.nth_root(5, all = True)
[]
sage: b.nth_root(3, all = True)
[14*a + 18, 10*a + 13, 5*a + 27]

sage: k.<a> = GF(29^5)
sage: b = a^2 + 5*a + 1
sage: b.nth_root(5)
19*a^4 + 2*a^3 + 2*a^2 + 16*a + 3
sage: b.nth_root(7)
Traceback (most recent call last):
...
ValueError: no nth root
sage: b.nth_root(4, all=True)
[]

Python

>>> from sage.all import *
>>> K = GF(Integer(31))
>>> a = K(Integer(22))
>>> K(Integer(22)).nth_root(Integer(7))
13
>>> K(Integer(25)).nth_root(Integer(5))
5
>>> K(Integer(23)).nth_root(Integer(3))
29

>>> K = GF(Integer(625), names=('a',)); (a,) = K._first_ngens(1)
>>> (Integer(3)*a**Integer(2)+a+Integer(1)).nth_root(Integer(13))**Integer(13)
3*a^2 + a + 1

>>> k = GF(Integer(29)**Integer(2), names=('a',)); (a,) = k._first_ngens(1)
>>> b = a**Integer(2) + Integer(5)*a + Integer(1)
>>> b.nth_root(Integer(11))
3*a + 20
>>> b.nth_root(Integer(5))
Traceback (most recent call last):
...
ValueError: no nth root
>>> b.nth_root(Integer(5), all = True)
[]
>>> b.nth_root(Integer(3), all = True)
[14*a + 18, 10*a + 13, 5*a + 27]

>>> k = GF(Integer(29)**Integer(5), names=('a',)); (a,) = k._first_ngens(1)
>>> b = a**Integer(2) + Integer(5)*a + Integer(1)
>>> b.nth_root(Integer(5))
19*a^4 + 2*a^3 + 2*a^2 + 16*a + 3
>>> b.nth_root(Integer(7))
Traceback (most recent call last):
...
ValueError: no nth root
>>> b.nth_root(Integer(4), all=True)
[]

ALGORITHM:

The default is currently an algorithm described in [Joh1999].

AUTHOR:

David Roe (2010-02-13)

pth_power(k=1)[source]¶

Return the \((p^k)\)-th power of self, where \(p\) is the characteristic of the field.

INPUT:

k – integer (default: 1, must fit in C int type)

Note that if \(k\) is negative, then this computes the appropriate root.

EXAMPLES:

Sage

sage: F.<a> = GF(29^2)
sage: z = a^2 + 5*a + 1
sage: z.pth_power()
19*a + 20
sage: z.pth_power(10)
10*a + 28
sage: z.pth_power(-10) == z
True
sage: F.<b> = GF(2^12)
sage: y = b^3 + b + 1
sage: y == (y.pth_power(-3))^(2^3)
True
sage: y.pth_power(2)
b^7 + b^6 + b^5 + b^4 + b^3 + b

Python

>>> from sage.all import *
>>> F = GF(Integer(29)**Integer(2), names=('a',)); (a,) = F._first_ngens(1)
>>> z = a**Integer(2) + Integer(5)*a + Integer(1)
>>> z.pth_power()
19*a + 20
>>> z.pth_power(Integer(10))
10*a + 28
>>> z.pth_power(-Integer(10)) == z
True
>>> F = GF(Integer(2)**Integer(12), names=('b',)); (b,) = F._first_ngens(1)
>>> y = b**Integer(3) + b + Integer(1)
>>> y == (y.pth_power(-Integer(3)))**(Integer(2)**Integer(3))
True
>>> y.pth_power(Integer(2))
b^7 + b^6 + b^5 + b^4 + b^3 + b

pth_root(k=1)[source]¶

Return the \((p^k)\)-th root of self, where \(p\) is the characteristic of the field.

INPUT:

k – integer (default: 1, must fit in C int type)

Note that if \(k\) is negative, then this computes the appropriate power.

EXAMPLES:

Sage

sage: F.<b> = GF(2^12)
sage: y = b^3 + b + 1
sage: y == (y.pth_root(3))^(2^3)
True
sage: y.pth_root(2)
b^11 + b^10 + b^9 + b^7 + b^5 + b^4 + b^2 + b

Python

>>> from sage.all import *
>>> F = GF(Integer(2)**Integer(12), names=('b',)); (b,) = F._first_ngens(1)
>>> y = b**Integer(3) + b + Integer(1)
>>> y == (y.pth_root(Integer(3)))**(Integer(2)**Integer(3))
True
>>> y.pth_root(Integer(2))
b^11 + b^10 + b^9 + b^7 + b^5 + b^4 + b^2 + b

sqrt(extend=False, all=False)[source]¶

See square_root().

EXAMPLES:

Sage

sage: k.<a> = GF(3^17)
sage: (a^3 - a - 1).sqrt()
a^16 + 2*a^15 + a^13 + 2*a^12 + a^10 + 2*a^9 + 2*a^8 + a^7 + a^6 + 2*a^5 + a^4 + 2*a^2 + 2*a + 2

Python

>>> from sage.all import *
>>> k = GF(Integer(3)**Integer(17), names=('a',)); (a,) = k._first_ngens(1)
>>> (a**Integer(3) - a - Integer(1)).sqrt()
a^16 + 2*a^15 + a^13 + 2*a^12 + a^10 + 2*a^9 + 2*a^8 + a^7 + a^6 + 2*a^5 + a^4 + 2*a^2 + 2*a + 2

square_root(extend=False, all=False)[source]¶

The square root function.

INPUT:

extend – boolean (default: True); if True, return a square root in an extension ring, if necessary. Otherwise, raise a ValueError if the root is not in the base ring.

Warning

This option is not implemented!
all – boolean (default: False); if True, return all square roots of self, instead of just one

Warning

The 'extend' option is not implemented (yet).

EXAMPLES:

Sage

sage: F = FiniteField(7^2, 'a')
sage: F(2).square_root()
4
sage: F(3).square_root()
2*a + 6
sage: F(3).square_root()**2
3
sage: F(4).square_root()
2
sage: K = FiniteField(7^3, 'alpha', impl='pari_ffelt')
sage: K(3).square_root()
Traceback (most recent call last):
...
ValueError: must be a perfect square.

Python

>>> from sage.all import *
>>> F = FiniteField(Integer(7)**Integer(2), 'a')
>>> F(Integer(2)).square_root()
4
>>> F(Integer(3)).square_root()
2*a + 6
>>> F(Integer(3)).square_root()**Integer(2)
3
>>> F(Integer(4)).square_root()
2
>>> K = FiniteField(Integer(7)**Integer(3), 'alpha', impl='pari_ffelt')
>>> K(Integer(3)).square_root()
Traceback (most recent call last):
...
ValueError: must be a perfect square.

to_bytes(byteorder='big')[source]¶

Return an array of bytes representing an integer.

Internally relies on the python int.to_bytes() method. Length of byte array is determined from the field’s order.

INPUT:

byteorder – string (default: 'big'); determines the byte order of the output; can only be 'big' or 'little'

EXAMPLES:

Sage

sage: F.<z5> = GF(3^5)
sage: a = z5^4 + 2*z5^3 + 1
sage: a.to_bytes()
b'\x88'

Python

>>> from sage.all import *
>>> F = GF(Integer(3)**Integer(5), names=('z5',)); (z5,) = F._first_ngens(1)
>>> a = z5**Integer(4) + Integer(2)*z5**Integer(3) + Integer(1)
>>> a.to_bytes()
b'\x88'

Sage

sage: F.<z3> = GF(163^3)
sage: a = 136*z3^2 + 10*z3 + 125
sage: a.to_bytes()
b'7)\xa3'

Python

>>> from sage.all import *
>>> F = GF(Integer(163)**Integer(3), names=('z3',)); (z3,) = F._first_ngens(1)
>>> a = Integer(136)*z3**Integer(2) + Integer(10)*z3 + Integer(125)
>>> a.to_bytes()
b'7)\xa3'

to_integer(reverse=False)[source]¶

Return an integer representation of this finite field element obtained by lifting its representative polynomial to \(\ZZ\) and evaluating it at the characteristic \(p\).

If reverse is set to True (default: False), the list of coefficients is reversed prior to evaluation.

Inverse of sage.rings.finite_rings.finite_field_base.FiniteField.from_integer().

EXAMPLES:

Sage

sage: F.<t> = GF(7^5)
sage: F(5).to_integer()
5
sage: t.to_integer()
7
sage: (t^2).to_integer()
49
sage: (t^2+1).to_integer()
50
sage: (t^2+t+1).to_integer()
57

Python

>>> from sage.all import *
>>> F = GF(Integer(7)**Integer(5), names=('t',)); (t,) = F._first_ngens(1)
>>> F(Integer(5)).to_integer()
5
>>> t.to_integer()
7
>>> (t**Integer(2)).to_integer()
49
>>> (t**Integer(2)+Integer(1)).to_integer()
50
>>> (t**Integer(2)+t+Integer(1)).to_integer()
57

Sage

sage: F.<t> = GF(2^8)
sage: u = F.from_integer(0xd1)
sage: bin(u.to_integer(False))
'0b11010001'
sage: bin(u.to_integer(True))
'0b10001011'

Python

>>> from sage.all import *
>>> F = GF(Integer(2)**Integer(8), names=('t',)); (t,) = F._first_ngens(1)
>>> u = F.from_integer(Integer(0xd1))
>>> bin(u.to_integer(False))
'0b11010001'
>>> bin(u.to_integer(True))
'0b10001011'

trace()[source]¶

Return the trace of this element, which is the sum of the Galois conjugates.

EXAMPLES:

Sage

sage: S.<a> = GF(5^3); S
Finite Field in a of size 5^3
sage: a.trace()
0
sage: a.charpoly('t')
t^3 + 3*t + 3
sage: a + a^5 + a^25
0
sage: z = a^2 + a + 1
sage: z.trace()
2
sage: z.charpoly('t')
t^3 + 3*t^2 + 2*t + 2
sage: z + z^5 + z^25
2

Python

>>> from sage.all import *
>>> S = GF(Integer(5)**Integer(3), names=('a',)); (a,) = S._first_ngens(1); S
Finite Field in a of size 5^3
>>> a.trace()
0
>>> a.charpoly('t')
t^3 + 3*t + 3
>>> a + a**Integer(5) + a**Integer(25)
0
>>> z = a**Integer(2) + a + Integer(1)
>>> z.trace()
2
>>> z.charpoly('t')
t^3 + 3*t^2 + 2*t + 2
>>> z + z**Integer(5) + z**Integer(25)
2

class sage.rings.finite_rings.element_base.FiniteRingElement[source]¶

Bases: CommutativeRingElement

to_bytes(byteorder='big')[source]¶

Return an array of bytes representing an integer.

Internally relies on the python int.to_bytes() method. Length of byte array is determined from the field’s order.

INPUT:

byteorder – string (default: 'big'); determines the byte order of input_bytes; can only be 'big' or 'little'

EXAMPLES:

Sage

sage: F = GF(65537)
sage: a = F(8726)
sage: a.to_bytes()
b'\x00"\x16'
sage: a.to_bytes(byteorder='little')
b'\x16"\x00'

Python

>>> from sage.all import *
>>> F = GF(Integer(65537))
>>> a = F(Integer(8726))
>>> a.to_bytes()
b'\x00"\x16'
>>> a.to_bytes(byteorder='little')
b'\x16"\x00'

sage.rings.finite_rings.element_base.is_FiniteFieldElement(x)[source]¶

Return True if x is a finite field element.

This function is deprecated.

EXAMPLES:

Sage

sage: from sage.rings.finite_rings.element_base import is_FiniteFieldElement
sage: is_FiniteFieldElement(1)
doctest:...: DeprecationWarning: the function is_FiniteFieldElement is deprecated; use isinstance(x, sage.structure.element.FieldElement) and x.parent().is_finite() instead
See https://github.com/sagemath/sage/issues/32664 for details.
False
sage: is_FiniteFieldElement(IntegerRing())
False
sage: is_FiniteFieldElement(GF(5)(2))
True

Python

>>> from sage.all import *
>>> from sage.rings.finite_rings.element_base import is_FiniteFieldElement
>>> is_FiniteFieldElement(Integer(1))
doctest:...: DeprecationWarning: the function is_FiniteFieldElement is deprecated; use isinstance(x, sage.structure.element.FieldElement) and x.parent().is_finite() instead
See https://github.com/sagemath/sage/issues/32664 for details.
False
>>> is_FiniteFieldElement(IntegerRing())
False
>>> is_FiniteFieldElement(GF(Integer(5))(Integer(2)))
True