# Description

In **Algebra**, one deals with the solutions of polynomial equations in one variable X, such as X^{2} - 2 = 0 or X^{5} + 4·X + 2 = 0. The set of all solutions of such polynomial equations with rational numbers as coefficients forms one field **Q**^{alg}. This means that you can add, subtract, multiply, and divide the numbers in it. One of the main problems of **Number Theory** is to better understand the field **Q**^{alg}. On the other hand, if polynomial equations (or systems of polynomial equations) are considered in several variables, their sets of solutions have a geometric structure (see the picture below). Also, e.g. the real solution set of X^{2} + Y^{2} - 1 = 0 is exactly the circle line. In addition, these sets of solutions also have a rich algebraic structure that results from considering polynomial equations. The branch of mathematics that investigates these sets of solutions is called **Algebraic Geometry**. A more detailed description can be found here.

"**Zitrus**" by Herwig Hauser is the real solution set of X^{2} + Z^{2} + Y^{3}·(Y−1)^{3} = 0.

(Own work by User: Andreasmatt. Licensed under Creative Commons Attribution-Share Alike 3.0 via Wikimedia Commons https://commons.wikimedia.org/wiki/File:IMAGINARY_Zitrus_Herwig_Hauser.jpg)

Other algebraic areas that are the focus in Münster are **Representation Theory**, **Group Theory** and the **Theory of Automorphic Forms**. Of course, algebra is not an isolated individual science, but is closely related to other areas of mathematics (such as differential geometry, topology, logic, and operator algebras).