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]
>>> 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
>>> 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
>>> 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
>>> 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
\(p\)-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: 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)
>>> 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 \(GF(97)[x]/(x^3+7)\), and demonstrate many basic functions with it.
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
>>> 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 \(x\) 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: 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)
>>> 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)