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2.11.1 Introduction to RINGELEM
An object of type RINGELEM in CoCoA represents an element of a ring.

To fix terminology about polynomials (elements of a polynomial ring): a polynomial is a sum of terms; each term is the product of a coefficient and power-product, a power-product being a product of powers of indeterminates. (In English it is standard to use monomial to mean a power-product, however, in other languages, such as Italian, monomial connotes a power-product multiplied by a scalar. In the interest of world peace, we will use the term power-product in those cases where confusion may arise.)

Example
/**/  use P ::= QQ[x,y,z];
/**/  f := 3*x*y*z + x*y^2;
/**/  f;
x*y^2 + 3*x*y*z
-------------------------------
/**/  use P ::= QQ[x[1..5]];
/**/  sum([x[N]^2 | N in 1..5]);
x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 + x[5]^2
-------------------------------
CoCoA always keeps polynomials ordered with respect to the term-orderings of their corresponding rings.

The following algebraic operations on polynomials are supported:
  F^N, +F, -F, F*G, F/G if G divides F, F+G, F-G,
where F, G are polynomials and N is an integer. The result may be a rational function.

Example
  Use R ::= QQ[x,y,z];
  F := x^2+xy;
  G := x;
  F/G;
x + y
-------------------------------
  F/(x+z);
(x^2 + xy)/(x + z)
-------------------------------
  F^2;
x^4 + 2x^3y + x^2y^2
-------------------------------
  F^(-1);
1/(x^2 + xy)
-------------------------------