Albion College Mathematics and Computer Science Colloquium

Title: Abel's Impossibility Theorem

Speaker: Susan J. Sierra
Graduate Student
Mathematics
University of Michigan
Ann Arbor, Michigan

Abstract:
You know the quadratic formula, but what about the cubic formula: if
$\begin{displaymath}x^3 + px + q = 0\end{displaymath}$

then
$\begin{displaymath}x = \omega^i \sqrt[3]{ â \frac{q}{2} + \sqrt{\left( \frac{q......{\left( \frac{q}{2} \right)^2 + \left( \frac{p}{3}\right)^3}}\end{displaymath}$

(for = 0, 1, or 2). Imagine having to memorize that for an exam!
There's also a quartic formula for fourth degree equations. You may have heard, however, that there is no formula to solve a quintic polynomial by adding, subtracting, multiplying, dividing, and taking roots of the coefficients. This was proved by the great Norwegian mathematican Niels Henrik Abel in 1824.
We'll talk about the elegant algebraic structures that encode information about solving polynomials, do a bit of basic group theory and Galois theory, and prove Abel's "impossibility theorem." Time permitting, we'll end with some intriguing mathematical puzzles.

Location: Palenske 227

Date: 11/1/2007

Time: 3:10 PM

@abstract{MCS:Colloquium:SusanJSierra
GraduateStudent
Mathematics
UniversityofMichigan
AnnArborMichigan:2007:11:1, author = "{Susan J. Sierra
Graduate Student
Mathematics
University of Michigan
Ann Arbor, Michigan}", title = "{Abel's Impossibility Theorem}", address = "{Albion College Mathematics and Computer Science Colloquium}", month = "{1 November}", year = "{2007}" }

Title:	Abel's Impossibility Theorem
Speaker:	Susan J. Sierra Graduate Student Mathematics University of Michigan Ann Arbor, Michigan
Abstract:	You know the quadratic formula, but what about the cubic formula: if $\begin{displaymath}x^3 + px + q = 0\end{displaymath}$ then $\begin{displaymath}x = \omega^i \sqrt[3]{ â \frac{q}{2} + \sqrt{\left( \frac{q......{\left( \frac{q}{2} \right)^2 + \left( \frac{p}{3}\right)^3}}\end{displaymath}$ (for = 0, 1, or 2). Imagine having to memorize that for an exam! There's also a quartic formula for fourth degree equations. You may have heard, however, that there is no formula to solve a quintic polynomial by adding, subtracting, multiplying, dividing, and taking roots of the coefficients. This was proved by the great Norwegian mathematican Niels Henrik Abel in 1824. We'll talk about the elegant algebraic structures that encode information about solving polynomials, do a bit of basic group theory and Galois theory, and prove Abel's "impossibility theorem." Time permitting, we'll end with some intriguing mathematical puzzles.
Location:	Palenske 227
Date:	11/1/2007
Time:	3:10 PM