Dear colleagues,

You are most cordially invited to the first of the Yeditepe Mathematics Department 25th Year Seminars of the current semester, organized by the Department of Mathematics. All students are also welcome. The details of this week's talk are as follows.

**Title: **The impossibility of the angle trisection by straightedge and compass revisited

**Speaker:** Yusuf Ünlü (Yeditepe Üniversitesi)

**Abstract:** It is well known that the angle trisection or doubling the cube is impossible by using only ruler and compass. The usual algebraic proof uses the fact that if a real number $\xi$ is constructible using only ruler and compass then there is a tower of fields $$\mathbb Q= F_0 \subset F_1 \subset \cdots \subset F_n$$ such that if $n \geq 1$, then $F_i = F_{i-1}(u_i)$ where $u_i\notin F_{i-1}$ but $u_i^2\in F_{i-1}$ and $\xi \in F_n$. Hence $2^m = [F_n : \mathbb Q]$ for some $m \in \mathbb N$. This shows that $$2^m = [F_n : \mathbb Q]= [F_n : \mathbb Q(\xi)][\mathbb Q(\xi) : \mathbb Q]$$ So, if the minimal polynomial of $\xi$ in $\mathbb Q(x)$ is of odd degree, then $\xi$ is not constructible by only ruler and compass. Moreover, the minimal polynomial of $\cos 20^\circ$ has degree $3$. So it is impossible to trisect $60^\circ$ by using only ruler and compass.

However, this proof requires the fundamentals of vector spaces. In this talk, we will tweak the last part of the proof to avoid vector spaces.

**Date:** Friday, October 1, 2021

**Time:** 16:00

**Place:** Contact Asst. Prof. Dr. Mehmet Akif ERDAL (mehmet.erdal@yeditepe.edu.tr).

You can find a copy of the flyer in this link. *For a full list of this semester's seminar schedule, click on the link below!*

https://researchseminars.org/seminar/7tepemathseminars

Zoom: Contact Asst. Prof. Dr. Mehmet Akif ERDAL (mehmet.erdal@yeditepe.edu.tr) for the address.