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?!

Why Is Many-One Reducibility Not A Total Order?

We use the notation $A \leq_m B$ if there’s a many-one reduction from set $A$ to set $B$. When I first learned about many-one reduction, the notation $\leq_m$ made me think $\leq_m$ imposes a total order among subsets of $\mathbb{N}$, i.e. $\leq_m$ on $\mathcal{P}(\mathbb{N})$ behaves like $\leq$ on $\mathbb{N}$. Unfortunately, $\leq_m$ only imposes a partial order on $\mathcal{P}(\mathbb{N})$. In this post, we will discuss an example of two subsets of natural numbers that are not comparable using $\leq_m$.

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.

Random Coin

Let’s play a game. I toss a coin twice and ask you, what is the probability of landing two heads? I claim that my coin is a fair coin. Considering I might be lying, what probability should you guess?