Algebraic Geometry/Introduction

Algebraic Geometry : Introduction

What is Algebraic Geometry?
Algebraic Geometry is a branch of mathematics that combines abstract algebra with geometry - more precisely; it is the study of algebraic objects using geometrical tools. It can be seen as a combination of linear algebra ("systems of linear equations in several variables"), and algebra ("study of polynomial equations in one variable" (though not exclusively)). Perhaps another description is that algebraic geometry is the study of polynomial functions and the spaces on which those polynomial functions are defined (known as algebraic varieties).

Therefore, the starting point for algebraic geometry is the study of the solutions of systems of polynomial equations:

$$f_i(X_1,X_2,...,X_n) = 0, i = 1,2,...,m, f_i \in k[X_1,X_2,...,X_n]$$

The theorems for polynomial equations depend on whether or not the field $$k$$ is algebraically closed and whether $$k$$ has characteristic 0.

For example, let $$k = \mathbb{Q}$$ and $$f (X_1,X_2,X_3) = X_1^n+X_2^n-X_3^n $$. This is Fermat's Last Theorem - and if $$n \ge 3$$ then only trivial solutions to the problem exist.

What questions can be asked of this system of polynomial equations? It is not possible in many circumstances to note all the solutions explicitly (remember that a single polynomial equation cannot be solved in general exactly if it's degree is greater than 4 - see Abel–Ruffini theorem) - so most study is dedicated towards the geometrical structure of the set of solutions to these equations.

Notation
Throughout the book, a ring will generally be taken as a commutative ring with identity, and $$k$$ will represent an algebraically closed field.

Preliminaries
In order to progress in the subject, we must review some results from commutative algebra (i.e. the study of commutative rings).

History
The field of algebraic geometry was first developed by Islamic mathematicians, such as the Persian mathematician Omar Khayyám.