Algebra 2 (2023) Course Notes
Table of Contents
Disclaimer: These notes may not be complete, accurate, or accurately reflect what happened in class.
1. Introduction
Mathematics is broadly about two things: numbers and shapes. Of these, numbers are plainly in the realm of Algebra. We can also bring shapes into the realm of Algebra by considering functions on the shapes. Numbers and functions can be added and multiplied. More precisely, they form rings. In this class, we will study numbers and functions through the language of rings and fields.
1.1. Example: Integers and rational numbers
The simplest kinds of numbers are whole numbers, or elements of the ring \(\mathbf{Z}\). The next simplest are quotients of integers, or elements of the ring \(\mathbf{Q}\). Most of our “number rings” will be extensions of these.
1.2. Example: Polynomials and rational functions
The simplest kinds of functions are polynomial functions, or elements of the ring \(\mathbf{R}[x]\). (You can replace \(\mathbf{R}\) by others, like \(\mathbf{C}\) or \(\mathbf{Q}\)). The next simplest are the rational functions, or elements of \(\mathbf{R}(x)\). Most of our “function rings” will be extensions of these.
1.3. Example: Finite fields
There are other extremely important rings related to the integers. These are the rings \(\mathbf{Z}/n \mathbf{Z}\). We will mostly focus on the case of \(n = p\) a prime number, so that the ring above is a field. We will meet extensions of these, which are strictly speaking neither “function rings” or “number rings”, and not strictly speaking are both.
1.4. More number rings
Fix a number \(\alpha \in \mathbf{C}\). We denote by \(\mathbf{Z}[\alpha] \subset \mathbf{C}\) the smallest sub-ring of \(\mathbf{C}\) containing \(\mathbf{Z}\) and \(\alpha\). In other words, see for yourself that \[ \mathbf{Z}[\alpha] = \{a_{0} + a_1 \alpha + \dots + a_{n}\alpha^{n} \mid a_i \in \mathbf{Z}\}.\] More generally, given \(\alpha_{1}, \dots, \alpha_{m} \in \mathbf{C}\), we denote by \(\mathbf{Z}[\alpha_{1}, \dots, \alpha_{m}] \subset \mathbf{C}\) the smallest subring of \(\mathbf{C}\) containing \(\mathbf{Z}\) and \(\alpha_1, \dots, \alpha_{m}\). We can write elements of \(\mathbf{Z}[\alpha_{1},\dots, \alpha_{m}]\) just as before. More precisely, we have the following observation.
1.4.1. Proposition
Let \(\alpha_{1}, \dots, \alpha_{m} \in \mathbf{C}\) be given. Then \(\mathbf{Z}[\alpha_{1}, \dots, \alpha_{m}] \subset \mathbf{C}\) is the image of the homomorphism \[ \mathbf{Z}[x_1, \dots, x_m] \to \mathbf{C}\] that sends \(x_i \to \alpha_{i}\).
1.4.2. Example
Let us take \(\alpha = i\). Then you will see that \(\mathbf{Z}[i]\) can be described more simply by \[ \mathbf{Z}[i] = \{a_{0} + a_{1} i \mid a_{0}, a_{1} \in \mathbf{Z}\}.\] Higher powers of \(\alpha\) are not necessary!
1.4.3. Example
Take \(\alpha = \sqrt[3]{2}\) and notice that we only need the 0th, 1st, and 2nd power of \(\alpha\).
1.4.4. Example
Take \(\alpha = \pi\). Then, it is not clear that we can get away with only finitely many powers!
1.4.5. Lesson
The examples above suggest that sometimes, we can describe \(\mathbf{Z}[\alpha]\) using finitely many powers of \(\alpha\) and sometimes we cannot. This is an important distinction that we will formalise later.
1.5. Generalisation of \(\mathbf{Z}[\alpha_{1}, \dots, \alpha_{m}].\)
Let \(R \subset S\) be a sub-ring. Given \(\alpha_{1}, \dots, \alpha_{m}\), we define \(R[\alpha_{1}, \dots, \alpha_{m}]\) as the smallest subring of \(S\) that contains \(R\) and each of \(\alpha_{1}, \dots, \alpha_{m}\). As before, it is the image of the homomorphism \[ R[x_{1}, \dots, x_{n}] \to S\] that sends \(R \to S\) by the given inclusion and sends \(x_{i}\) to \(\alpha_{i}\).