# Elisabeth Bonnevier: An introduction to algebraic geometry and Bezout's theorem.

**Time: **
Wed 2018-06-13 11.00 - 12.00

**Location: ** Room 22, House 5, Kräftriket, Department of Mathematics, Stockholm University

**Doctoral student: **
Elisabeth Bonnevier (BSc student)

**Supervisor: **
Gregory Arone

Abstract:

*The fundamental theorem of algebra tells us the number of roots of a **polynomial. As a generalization, Bezout's theorem tells us the number of **intersection points between two arbitrary polynomial curves in a plane. *

*The aim of this text is to develop some of the theory of algebraic **geometry and prove Bezout's theorem. First, after some initial **definitions and propositions we will prove the classical result of **Hilbert's nullstellensatz, which describes the relationship between **algebraic sets and ideals of a polynomial ring. From that we continue on **to define the projective space, to which we extend our previous **definitions of algebraic sets and ideals. Also needed for Bezout's **theorem is the notion of intersection number, which is a generalization **of counting zeros with multiplicities. The properties expected of the **intersection number are given and we show that there is only one number *

*which satisfies those properties. Then we have all the theory needed and **we will prove Bezout's theorem.*