Algebraic geometry as the names says, is the concept of geometry in algebra. In other words, the relation of algebra with its geometrical representation is known as algebraic geometry. It is basically, the algebra of rings, especially the ring of polynomials, (where a ring is denoted by a non empty set R, of real numbers satisfying a list of properties) and their graphical representation of the zeroes of the polynomials.
The formal definition of Ring is as follows:
A ring is a non empty set R with two binary compositions, denoted by ' + ' and ' * ' respectively, satisfying the following properties:
1. a + b belongs to R, for all a, b belonging to R.
2. a + b = b + a, for all a, b belonging to R.
3. a + (b + c) = (a + b) + c, for all a, b, c belonging to R.
4. Property of additive identity: There exists an element denoted by 0 belonging to R, such that: a + 0 = a, for all a belonging to R.
Here, 0 is called as an additive identity or zero element in R.
5. Property of additive inverse: For each a belonging to R, there exist an element b belonging to R such that: a + b = 0,
Here, b is called an additive inverse or negative of a, and is written as b = -a, so that a + (-a) = 0.
6. a, b belongs to R, for all a, b belonging to R.
7. a * (b * c) = (a * b) * c, for all a, b, c, belonging to R.
8. For all a, b, c belonging to R,
- a * (b + c) = a * b + a * c, which is known as the left distributive law.
- (b + c) * a = b * a + c * a, which is known as the right distributive law.
If the properties of 1 to 5, is satisfied then we say [R, +] is an Abelian group.
A ring [R, +, *] is called a commutative ring if a * b = b * a for all a, b belonging to R.
A ring [R, +, *] is called a field if it’s non zero elements form an Abelian group with respect to the composition ‘*’ i.e. if R is a field, then 'a' belonging to R, where 'a' not equal to zero, implies a-1 belongs to R.
A polynomial ring is denoted as R [x], in one variable over a commutative ring R.Once we know the ring of polynomials we can easily relate it to the 2-D or 3-D or in any number of fields. For example, any linear polynomial in first order will always represent a straight line in a plane, as shown below: