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Oscar Xu's review notes
A brief review of key concepts in Math 104. The notes are organized in the order of chapters with respect to Professor Peng Zhou's lectures.
Chapter 1: Introduction
$\mathbb{Z}$: integers, has subtraction and zero. $\mathbb{Z}$ is an example of “ring”.
$\mathbb{Q}$: rational number, ($\frac{n}{m}, n,m \in \mathbb{Z}$)
What is the root of $x^2 - 2 = 0$?
Definition of algebraic numbers: A number is called an algebraic number if it satisfies a polynomial equation
$c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0 = 0$
Where the coefficients $c_0, c_1, …, c_n$ are integers,
Rational Zeros Theorem: Supose $c_0, c_1, \cdots, c_n$ are integers and $r$ is a rational number satisfying the polynomial equation
$c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0 = 0$
where $n \geq 1$, $c_n \neq 0$, $c_0 \neq 0$. Let $r = \frac{c}{d}$, where c, d are relatively prime, then $c$ divides $c_0$ and d divides $c_n$
