Rings#
Matrix rings#
How do you construct a matrix ring over a finite ring in Sage? The
MatrixSpace constructor accepts any ring as a base ring. Here’s
an example of the syntax:
sage: R = IntegerModRing(51)
sage: M = MatrixSpace(R,3,3)
sage: M(0)
[0 0 0]
[0 0 0]
[0 0 0]
sage: M(1)
[1 0 0]
[0 1 0]
[0 0 1]
sage: 5*M(1)
[5 0 0]
[0 5 0]
[0 0 5]
R = IntegerModRing(51) M = MatrixSpace(R,3,3) M(0) M(1) 5*M(1)
>>> from sage.all import *
>>> R = IntegerModRing(Integer(51))
>>> M = MatrixSpace(R,Integer(3),Integer(3))
>>> M(Integer(0))
[0 0 0]
[0 0 0]
[0 0 0]
>>> M(Integer(1))
[1 0 0]
[0 1 0]
[0 0 1]
>>> Integer(5)*M(Integer(1))
[5 0 0]
[0 5 0]
[0 0 5]
Polynomial rings#
How do you construct a polynomial ring over a finite field in Sage? Here’s an example:
sage: R = PolynomialRing(GF(97),'x')
sage: x = R.gen()
sage: f = x^2+7
sage: f in R
True
R = PolynomialRing(GF(97),'x') x = R.gen() f = x^2+7 f in R
>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(97)),'x')
>>> x = R.gen()
>>> f = x**Integer(2)+Integer(7)
>>> f in R
True
Here’s an example using the Singular interface:
sage: R = singular.ring(97, '(a,b,c,d)', 'lp')
sage: I = singular.ideal(['a+b+c+d', 'ab+ad+bc+cd', 'abc+abd+acd+bcd', 'abcd-1'])
sage: R
polynomial ring, over a field, global ordering
// coefficients: ZZ/97
// number of vars : 4
// block 1 : ordering lp
// : names a b c d
// block 2 : ordering C
sage: I
a+b+c+d,
a*b+a*d+b*c+c*d,
a*b*c+a*b*d+a*c*d+b*c*d,
a*b*c*d-1
R = singular.ring(97, '(a,b,c,d)', 'lp') I = singular.ideal(['a+b+c+d', 'ab+ad+bc+cd', 'abc+abd+acd+bcd', 'abcd-1']) R I
>>> from sage.all import *
>>> R = singular.ring(Integer(97), '(a,b,c,d)', 'lp')
>>> I = singular.ideal(['a+b+c+d', 'ab+ad+bc+cd', 'abc+abd+acd+bcd', 'abcd-1'])
>>> R
polynomial ring, over a field, global ordering
// coefficients: ZZ/97
// number of vars : 4
// block 1 : ordering lp
// : names a b c d
// block 2 : ordering C
>>> I
a+b+c+d,
a*b+a*d+b*c+c*d,
a*b*c+a*b*d+a*c*d+b*c*d,
a*b*c*d-1
Here is another approach using GAP:
sage: R = gap.new("PolynomialRing(GF(97), 4)"); R
PolynomialRing( GF(97), ["x_1", "x_2", "x_3", "x_4"] )
sage: I = R.IndeterminatesOfPolynomialRing(); I
[ x_1, x_2, x_3, x_4 ]
sage: vars = (I.name(), I.name(), I.name(), I.name())
sage: _ = gap.eval(
....: "x_0 := %s[1];; x_1 := %s[2];; x_2 := %s[3];;x_3 := %s[4];;"
....: % vars)
sage: f = gap.new("x_1*x_2+x_3"); f
x_2*x_3+x_4
sage: f.Value(I,[1,1,1,1])
Z(97)^34
R = gap.new("PolynomialRing(GF(97), 4)"); R
I = R.IndeterminatesOfPolynomialRing(); I
vars = (I.name(), I.name(), I.name(), I.name())
_ = gap.eval(
"x_0 := %s[1];; x_1 := %s[2];; x_2 := %s[3];;x_3 := %s[4];;"
% vars)
f = gap.new("x_1*x_2+x_3"); f
f.Value(I,[1,1,1,1])
>>> from sage.all import *
>>> R = gap.new("PolynomialRing(GF(97), 4)"); R
PolynomialRing( GF(97), ["x_1", "x_2", "x_3", "x_4"] )
>>> I = R.IndeterminatesOfPolynomialRing(); I
[ x_1, x_2, x_3, x_4 ]
>>> vars = (I.name(), I.name(), I.name(), I.name())
>>> _ = gap.eval(
... "x_0 := %s[1];; x_1 := %s[2];; x_2 := %s[3];;x_3 := %s[4];;"
... % vars)
>>> f = gap.new("x_1*x_2+x_3"); f
x_2*x_3+x_4
>>> f.Value(I,[Integer(1),Integer(1),Integer(1),Integer(1)])
Z(97)^34
How do you construct
To compute the characteristic and residue class field of the ring
Zp of integers of Qp, use the syntax illustrated by the
following examples.
sage: K = Qp(3)
sage: K.residue_class_field()
Finite Field of size 3
sage: K.residue_characteristic()
3
sage: a = K(1); a
1 + O(3^20)
sage: 82*a
1 + 3^4 + O(3^20)
sage: 12*a
3 + 3^2 + O(3^21)
sage: a in K
True
sage: b = 82*a
sage: b^4
1 + 3^4 + 3^5 + 2*3^9 + 3^12 + 3^13 + 3^16 + O(3^20)
K = Qp(3) K.residue_class_field() K.residue_characteristic() a = K(1); a 82*a 12*a a in K b = 82*a b^4
>>> from sage.all import *
>>> K = Qp(Integer(3))
>>> K.residue_class_field()
Finite Field of size 3
>>> K.residue_characteristic()
3
>>> a = K(Integer(1)); a
1 + O(3^20)
>>> Integer(82)*a
1 + 3^4 + O(3^20)
>>> Integer(12)*a
3 + 3^2 + O(3^21)
>>> a in K
True
>>> b = Integer(82)*a
>>> b**Integer(4)
1 + 3^4 + 3^5 + 2*3^9 + 3^12 + 3^13 + 3^16 + O(3^20)
Quotient rings of polynomials#
How do you construct a quotient ring in Sage?
We create the quotient ring
sage: R = PolynomialRing(GF(97),'x')
sage: x = R.gen()
sage: S = R.quotient(x^3 + 7, 'a')
sage: a = S.gen()
sage: S
Univariate Quotient Polynomial Ring in a over Finite Field of size 97 with
modulus x^3 + 7
sage: S.is_field()
True
sage: a in S
True
sage: x in S
True
sage: S.polynomial_ring()
Univariate Polynomial Ring in x over Finite Field of size 97
sage: S.modulus()
x^3 + 7
sage: S.degree()
3
R = PolynomialRing(GF(97),'x') x = R.gen() S = R.quotient(x^3 + 7, 'a') a = S.gen() S S.is_field() a in S x in S S.polynomial_ring() S.modulus() S.degree()
>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(97)),'x')
>>> x = R.gen()
>>> S = R.quotient(x**Integer(3) + Integer(7), 'a')
>>> a = S.gen()
>>> S
Univariate Quotient Polynomial Ring in a over Finite Field of size 97 with
modulus x^3 + 7
>>> S.is_field()
True
>>> a in S
True
>>> x in S
True
>>> S.polynomial_ring()
Univariate Polynomial Ring in x over Finite Field of size 97
>>> S.modulus()
x^3 + 7
>>> S.degree()
3
In Sage, in means that there is a “canonical coercion” into the
ring. So the integer
You can also compute in quotient rings without actually computing
then using the command quo_rem as follows.
sage: R = PolynomialRing(GF(97),'x')
sage: x = R.gen()
sage: f = x^7+1
sage: (f^3).quo_rem(x^7-1)
(x^14 + 4*x^7 + 7, 8)
R = PolynomialRing(GF(97),'x') x = R.gen() f = x^7+1 (f^3).quo_rem(x^7-1)
>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(97)),'x')
>>> x = R.gen()
>>> f = x**Integer(7)+Integer(1)
>>> (f**Integer(3)).quo_rem(x**Integer(7)-Integer(1))
(x^14 + 4*x^7 + 7, 8)