Disclaimer: These are just notes for a lecture, not a polished writeup. In particular, references
are missing. Use at your own discretion.
In this lecture we will solve the membership problem for subgroups of permutation groups.
First some notation. We us
6.885 Algebra and Computation
October 3, 2005
Lecture 7
Lecturer: Madhu Sudan
Scribe: Jingbin Yin
Today we are going to continue our talk about factorization of polynomials over nite elds.
And then we will give a completely dierent deterministic algorithm
6.885 Algebra and Computation
October 5, 2005
Lecture 8
Lecturer: Madhu Sudan
Scribe: Guy Rothblum
Today we will complete the description of Berlekumps deterministic algorithm for
eciently factorizing polynomials over Fq (where q = pt for a prime p) in ti
6.885 Algebra and Computation
October 12, 2005
Lecture 9
Lecturer: Madhu Sudan
Scribe: Brendan Juba
Today, we will continue our approach to factoring bivariate polynomials. We will begin by recalling
Hensels Lifting Lemma, and we will discuss a few new th
6.885 Algebra and Computation
October 19, 2005
Lecture 11
Lecturer: Madhu Sudan
Scribe: Kyomin Jung
Today, we will give a denition of lattice in Rn and study its two specications, by primal basis and
dual basis. Then we will focus on the problem of nding
6.885 Algebra and Computation
October 17, 2005
Lecture 10
Lecturer: Madhu Sudan
1
Scribe: Ben Rossman
Clarication on Resultants
Consider polynomials f (x), g (x) R[x] where R is a unique factorization domain (with some notion of
order on its elements). In
6.885 Algebra and Computation
September 28, 2005
Lecture 6
Lecturer: Madhu Sudan
Scribe: Arnab Bhattacharyya
In the last lecture, we saw an algorithm to nd roots of a polynomial in a nite eld. In particular,
we noticed that if a polynomial f (x) Fq [x] ha
6.885 Algebra and Computation
September 26, 2005
Lecture 5
Lecturer: Madhu Sudan
Scribe: Elena Grigorescu
In todays lecture we will rst go through a brief response to the comments on the previous lecture.
The comments and the brief responses will be poste
6.885 Algebra and Computation
September 7, 2005
Lecture 1
Lecturer: Madhu Sudan
1
Scribe: Swastik Kopparty
Motivation
1.1
Historical
Historically, the rst algorithms were for manipulations of numbers. Take a simple example like that of
multiplying two n b
6.885 Algebra and Computation
September 12, 2004
Lecture 2
Lecturer: Madhu Sudan
Scribe: Joshua A. Grochow
This lecture begins a brief introduction to the algebraic structures we will be using throughout the
course groups, rings, and elds and some of thei
6.885 Algebra and Computation
September 14, 2005
Lecture 3
Lecturer: Madhu Sudan
1
Scribe: Victor Chen
Introduction
Today we will cover polynomial rings and look at the Division Algorithm and Gausss Lemma.
Then we will introduce nite elds.
2
Polynomial Ri
September 21, 2005
6.885 Algebra and Computation
Lecture 4
Lecturer: Madhu Sudan
Scribe: Karola Mszros
ea
Membership algorithm for the permutation group
In the last lecture we only sketched the proof of Gausss lemma.
Exercise 1. Prove Gausss lemma rigorou
6.885 Algebra and Computation
October 26, 2005
Lecture 13
Lecturer: Madhu Sudan
Scribe: Michael Manapat
In todays lecture, well complete the analysis of the LLL algorithm and nish factoring over Q[X]
(details will be left to the exercises).
1
LLL Analysis