These are some notes on error correcting codes. Two good sources for
this material are
From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
Sphere Packings, Lattices, and Simple Groups by J. H. Conway and N.
Sloane
The goal of these notes is to explain why any elliptic curve over C has
a Weierstrass uniformization, up to obvious changes of coordinates. These
notes are sketchy and they wade into topics that are beyond the scope of the
class invariance of domain, modu
The goal of these notes is to explain Weierstrass Uniformization.
1
Lattices
Say that 2 complex numbers and are independent of / is not real. For
instance 1 and i are independent.
A lattice in C is a set of points of the form
= cfw_m + n m, n Z ,
(1)
whe
The purpose of these notes is to explain the complex analysis you need
to know for the Weierstrass uniformization of elliptic curves over C .
1
A Resume of Results
Let U be an open set in C , the complex plane. Let f : U C be a
continuous map. We say that
The purpose of these notes is to prove a special case of the CayleyBacharach Theorem and then to prove Pascals Theorem as an application.
The main result we prove, the Grid Theorem, will be useful when we analyze
the group structure of an elliptic curve.
The purpose of these notes is to prove Lindemanns Theorem. The proof
is adapted from Jacobsons book Algebra I , but I simplied some of the
assumptions in order to make the proof easier. Also, I improved the proof
somewhat.
1
The Main Result
Here is Lindem
Math 154 Notes
These are some notes on solvability.
1: Roots of Unity Not Necessary: Let F be a eld of characteristic
zero. Let p(x) F [x] be a polynomial which is solvable by radicals. Herstein
proves that the Galois group of p(x) is solvable, assuming t
Constructing the 17-gon:
In these notes, Ill give a proof that the 17-gon is constructible. Ill also take
the opportunity to correct something that I didnt get right in the lecture.
Let F R denote the eld of numbers R such that there is a nite
tower of el
Math 154 Notes 2
In these notes Ill construct some explicit transcendental numbers. These
numbers are examples of Liouville numbers.
Let cfw_ak be a sequence of positive integers, and suppose that ak divides
ak+1 for all k . We say that cfw_ak has moder
Math 154 Notes 1
These are some notes on algebraic integers. Let C denote the complex numbers.
Denition 1: An algebraic integer is a number x C that satises an
integer monic polynomial. That is
xn + an1 xn1 + . + a1 x + a0 = 0;
a0 , ., an1 Z .
(1)
For ins