From Schmid.wiki

## Topics

Polynomial - Algebraic properties

## Vocabulary

- Field
- In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.

Examples:

- the real numbers
- the complex numbers

- Ring
- In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. A ring is a generalization of the integers. Other examples include the polynomials and the integers modulo n. The branch of abstract algebra which studies rings is called ring theory.

A ring is a set R equipped with two binary operations + : R × R → R and · : R × R → R, called addition and multiplication, such that:

* (R, +) is an abelian group with identity element 0: o (a + b) + c = a + (b + c) o 0 + a = a + 0 = a o a + b = b + a o For every a in R, there exists an element denoted −a, such that a + −a = −a + a = 0 * (R, ·) is a monoid with identity element 1: o (a·b)·c = a·(b·c) o 1·a = a·1 = a * Multiplication distributes over addition: o a·(b + c) = (a·b) + (a·c) o (a + b)·c = (a·c) + (b·c)

- Abelian Group
- In mathematics, an abelian group, also called a commutative group, is a group (G, * ) such that a * b = b * a for all a and b in G. In other words, the order in which the binary operation is performed doesn't matter. Such groups are generally easier to understand, although infinite abelian groups remain a subject of current research.

- Module over a ring
- an Abelian group over a ring G with a new operation
*scalar multiplication*which has the properties

- fixme

- Vector space
- a module over a field

- Monoid
- A monoid is a set M with binary operation * : M × M → M, obeying the following axioms:

* Associativity: for all a, b, c in M, (a*b)*c = a*(b*c) * Identity element: there exists an element e in M, such that for all a in M, a*e = e*a = a.

## References

- GNU Scientific Library is a C library for
**numerical computation**like finding roots of polynomials, compute FFT's, numerical integration, etc. - GiNaC extends C++ by a set of algebraic capabilities, such as symbolic differentiation, linear algebra, etc.