Enrichment • Series
Abstract Algebra and Number Theory.
BL
LS
Lucas S
This series ended on July 20, 2023. All 1:1 and group chats related to this series are disabled 7 days after the last session.
Series Details
About
This is a course on Abstract Algebra and Number Theory for the curious minds. Ideally learners enrolled in this series would have some experience with some basic proof writing, otherwise understanding the proofs will be tough. The goal here is to introduce learners into the vast world of both Abstract Algebra and Number Theory and most importantly, being able to see connections between the two subjects!
✋ ATTENDANCE POLICY
This series has lots of material to cover, so learners that happen to miss sessions will possibly fall behind. If this does happen then I strongly suggest to message me either privately or publicly so that I can either sort out a catch up session or direct learners to external materials.
Dates
July 3 - July 20
Learners
7 / 10
Total Sessions
16
About the Tutor
BL
LS
Hello! I am Lucas, and I am a university student studying pure math.
View Lucas S's Profile
Upcoming Sessions
0
Past Sessions
16
3
Jul
O
Session 1
Orientation
Let's get to know each other! We will also go over some basic proof writing for anyone that needs a reminder. Attendance here is not needed if you feel okay with proof writing!
4
Jul
O
Session 2
Orientation
Let's get to know each other! We will go over some basic proof writing for anyone that needs a reminder. Attendance is not needed here if you feel okay with proof writing!
5
Jul
OT
Session 3
Other Topics
We'll cover some basic facts about divisibility. In particular, we will learn how to precisely state what it means for two integers to divide into each other and some of the properties that come with it! After this, we will investigate a really important result known as the Division Algorithm. This allows us to generalise the idea of divisibility to have division with remainder! A big part of this session is to prove this algorithm. We'll also look at the Well-Ordering Principle and the greatest common divisor.
6
Jul
OT
Session 4
Other Topics
We will discover when a linear diophantine equation has a solution over the integers. This will quickly lead us into the Euclidean Algorithm and the Extended Euclidean Algorithm. We will also look at primes and some basic properties that primes have!
7
Jul
OT
Session 5
Other Topics
We will continue looking at properties that primes have. Another big milestone will be looking at the Fundamental Theorem of Arithmetic and its proof. Alongside this big theorem we will look at a result by Euclid from over 2000 years ago that there are infinitely many primes. At the end, we may get to look at the distribution of primes by looking at the Prime Number Theorem. We will not provide a proof here. The proof is well and truely outside the scope of this series!
8
Jul
OT
Session 6
Other Topics
We will continue our work on semi-groups and later move on to looking at groups! We will look at some basic examples of groups, followed by a nice result that will help us in our next bit of study on Number Theory. Afterwards, we dive back into Number Theory!
10
Jul
OT
Session 7
Other Topics
We will continue with investigating some Number Theory and see some of the connections it has with Algebra! We will also look at the Chinese Remainder Theorem.
11
Jul
R
Session 8
Review
We will provide a review of what we have done so far and answer any questions you may have! Note that this session is not important to attend if you are okay with the content so far.
12
Jul
OT
Session 9
Other Topics
We know look at the group of integers modulo n and the algebraic properties that this group has. From this we can define a new group, known as the group of units (or the multiplicative group of integers modulo n).
13
Jul
OT
Session 10
Other Topics
A big day, we look at the Chinese Remainder Theorem! This is a really big theorem in Number Theory. After this, we will move back to group theory looking at some more basic results to do with groups.
14
Jul
OT
Session 11
Other Topics
We will go through a quiz that covers the content that we have done so far! Hopefully this will be a useful review and also a fun activity to do. After this, we will move on to look at a special family of groups known as the Symmetric groups (along with another special class of groups known as the Dihedral Groups) and subgroups.
15
Jul
OT
Session 12
Other Topics
We will now look at the Dihedral Group and subgroups. The Dihedral Group will act as another example of an important type of group. We will also look at criterions for subgroups (the Subgroup Test).
16
Jul
OT
Session 13
Other Topics
We will look at what the order of an element is in a group and cyclic groups. This will prove useful for our later studies in Number Theory!
17
Jul
OT
Session 14
Other Topics
We will look at Group Homomorphisms and Group Isomorphisms. These are important maps between groups.
19
Jul
OT
Session 15
Other Topics
We will look back at some Number Theory. In particular, we will look at Euler's Totient Function and some classic theorems that come along with it!
20
Jul
OT
Session 16
Other Topics
This session will be a review of everything that we have covered so far. As well as a chance to go over the final quiz of the series! This is also the last session of the series.