# A Scary-Looking Quotient Ring …

I was preparing for my algebra final and I ran into this quotient ring constructed in $\mathbb{Z}[\sqrt{-5}]$:

$(1 + \sqrt{-5}, 3) / (1 + \sqrt{-5})$

What even is this?!

# Integral Domain

We covered Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains before Thanksgiving break. I was slightly stressed during the break because I didn’t grasp the concepts very well. Luckily, when I get stressed, I write notes, and then I’ll be happy again 😉 So here it is:

# Let’s Partition A Group!

I think of a group as a bag of items with a rule of smashing two items into one. Sometimes, we don’t want mix everything together in one bag; we might want to divide the items into several smaller bags. Let’s talk about two ways of partitioning a group, i.e. dividing the items in the bag.

# Shuffle, Shuffle, Shuffle …

Groups are like verbs. We can gain more information about a group by looking at how it interacts with other objects.

# What Does Order Tell You?

When I first learned algebra, I thought there’s no reason to believe we can gain structural information of a group solely by looking at the number of elements of the group. Unfortunately (or maybe fortunately :p), I was wrong.

# The Isomorphism Theorems

We learned about quotient groups and four Isomorphism Theorems in my algebra class. My professor refers the four Isomorphism Theorems as “four siblings of Isomorphism Theorems” (of course, that made me laugh; it always makes me happy when people describe math in lively terms). I’ll write about my understanding of them in this post.