March 12

Plan:

  • Questions
  • Kummer and the beginnings of algebraic number theory
  • Chapter summary

Student Questions: none


The discussion page can be found here, for observations and remarks.

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Build the Gaussian integers by adjoining $\sqrt{-1}$ to the integers. The resulting ring enjoys unique factorization. To get the definitions of "prime" and "uniqueness" right we need to talk about units and associates. This is moving pretty far from history toward real mathematics, but worth it.

Unique factorization fails for the adjunction of $\sqrt{-5}$.

Adjoining a primitive cube root of unity creates the Eisenstein integers. They do enjoy unique factorization. Factoring 4 there is interesting - it answers Matt's question at the end of the last class, but in a way that may feel unsatisfying.

Then on to cyclotomic fields, Kummer and the Fermat conjecture. Perhaps a detour into the Euclidean construction of regular polygons.


In today's class we summarized our work with Fermat's conjecture:
* $x^n +y^n \ne z^n$ Stated by Fermat in the 1600's
* Cases for 3, 4, 5, 7 are proved by other mathematicians such as Gauss, Germaine, and Euler
* Wiles proves this for all $n>2$ about 20 years ago
* In order to study and understand all of this we needed to first understand Pythagorean triples, basic number theory, and the Fundamental Theorem of Arithmetic

It was also noted that Kummer studied cases where the Fundamental Theorem of Arithmetic fails, this is similar to the way that another geometry exists where the parallel postulate is false.

We then discussed roots of unity and the complex plane
* In order to get roots of unity we need to have linear factors
* x{p - 1} = (x-1)(x{p-1}+…+ x+1)
* zp = xp - yp = (x-y) (x{p-1}y + x{p-2}y +…+ xy{p-1} + y These terms must also be squares

side note: if $ab = x^n$ where $a$ and $b$ are relatively prime then $a= u^n$ and $b= v^n$.

  • The norm of a complex number is its distance from the center of the complex plane. N(z) = length of z
  • N(zw) = N(z)N(w)

This discussion led to the question which regular n-gons could the Greeks construct using a straight-edge and compass
* The n-gon is constructable if n/2 can be constructed
* If you can build an n-gon and an m-gon where m and n are relatively prime then you can build an nm-gon
* The regular p-gon, where p is prime, can be constructed if and only if p = 2k+1 and k = 2n

Greek Geometers could not construct a heptagon with a ruler and compass.

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