# 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]
```

## 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
```

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
```

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
```

## \(p\)-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
folowing 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)
```

## 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
```

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)
```