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

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]

Python

>>> 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]

Sage Live

R = IntegerModRing(51)
M = MatrixSpace(R,3,3)
M(0)
M(1)
5*M(1)

Polynomial rings¶

How do you construct a polynomial ring over a finite field in Sage? Here’s an example:

Sage

sage: R = PolynomialRing(GF(97),'x')
sage: x = R.gen()
sage: f = x^2+7
sage: f in R
True

Python

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(97)),'x')
>>> x = R.gen()
>>> f = x**Integer(2)+Integer(7)
>>> f in R
True

Sage Live

R = PolynomialRing(GF(97),'x')
x = R.gen()
f = x^2+7
f in R

Here’s an example using the Singular interface:

Sage

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

Python

>>> 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

Sage Live

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

Here is another approach using GAP:

Sage

sage: R = libgap.PolynomialRing(GF(97), 4); R
GF(97)[x_1,x_2,x_3,x_4]
sage: I = R.IndeterminatesOfPolynomialRing(); I
[ x_1, x_2, x_3, x_4 ]
sage: x1, x2, x3, x4 = I
sage: f = x1*x2 + x3; f
x_1*x_2+x_3
sage: f.Value(I,[1,1,1,1])
Z(97)^34

Python

>>> from sage.all import *
>>> R = libgap.PolynomialRing(GF(Integer(97)), Integer(4)); R
GF(97)[x_1,x_2,x_3,x_4]
>>> I = R.IndeterminatesOfPolynomialRing(); I
[ x_1, x_2, x_3, x_4 ]
>>> x1, x2, x3, x4 = I
>>> f = x1*x2 + x3; f
x_1*x_2+x_3
>>> f.Value(I,[Integer(1),Integer(1),Integer(1),Integer(1)])
Z(97)^34

Sage Live

R = libgap.PolynomialRing(GF(97), 4); R
I = R.IndeterminatesOfPolynomialRing(); I
x1, x2, x3, x4 = I
f = x1*x2 + x3; f
f.Value(I,[1,1,1,1])

\(p\)-adic numbers¶

How do you construct \(p\)-adics in Sage? A great deal of progress has been made on this (see SageDays talks by David Harvey and David Roe). Here only a few of the simplest examples are given.

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

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)

Python

>>> 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)

Sage Live

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

Quotient rings of polynomials¶

How do you construct a quotient ring in Sage?

We create the quotient ring \(GF(97)[x]/(x^3+7)\), and demonstrate many basic functions with it.

Sage

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

Python

>>> 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

Sage Live

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()

In Sage, in means that there is a “canonical coercion” into the ring. So the integer \(x\) and \(a\) are both in \(S\), although \(x\) really needs to be coerced.

You can also compute in quotient rings without actually computing then using the command quo_rem as follows.

Sage

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)

Python

>>> 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)

Sage Live

R = PolynomialRing(GF(97),'x')
x = R.gen()
f = x^7+1
(f^3).quo_rem(x^7-1)