Rings and Polynomials

## Description

The basic rings in Sage are the integers (called ZZ), the rationals (called QQ), the reals (called RR), and the complex numbers (called CC). We can define polynomial rings over each of these fields; however, a polynomial defined over the rational numbers is different than a polynomial defined over the real numbers. For example, the polynomial \$p(x) = x^2 - 2\$ factors over the real numbers but not the rationals. The polynomial \$q(x) = x^2 + 2\$ factors over the complex numbers but not the real numbers. If we declare polynomial rings ratpoly.<t> = PolynomialRing(QQ), realpoly.<w> = PolynomialRing(RR), and complexpoly.<z> = PolynomialRing(CC), polynomials will factor differently in each case. The names ratpoly, realpoly, and complexpoly are not important, but the variables t, w, and z are important. The polynomial \$t^2 - 2\$ will not factor but the polynomial \$w^2 - 2\$ will.

## Sage Cell

#### Code

``````ratpoly.<t> = PolynomialRing(QQ)
factor(t^2-1)

ratpoly.<t> = PolynomialRing(QQ)
factor(t^2-2)```
```

## Options

#### Option

Factoring over the real numbers.

#### Code

``````realpoly.<w> = PolynomialRing(RR)
factor(w^2 - 2)```
```

#### Option

Factoring over the complex numbers. Note that I represents the square root of \$-1\$.

#### Code

``````complexpoly.<z> = PolynomialRing(CC)
factor(z^2 + 2)```
```

## Tags

Primary Tags: abstract algebra

Secondary Tags: rings, factoring, polynomials

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## Attribute

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Author: T. W. Judson

Date: 01 Aug 2018 00:17

Submitted by: Tom Judson

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