02 - Field

Field $\mathbf{F}$

Let $\mathbf{F}$ be a set with operations $+$ and $\cdot$ that satisfy the axioms
Then

$$\mathbf{F}\text{ is a field}$$

Example of Fields

1. $\mathbb{R}$ - Real numbers
2. $\mathbb{Q}$ - Rational numbers
3. $\mathbb{C}$ - Complex numbers

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Notation on operations for arithmetic

$$\begin{array}{rl}a-b& \text{ for }a+(-b)\\ ab& \text{ for }a\cdot b\\ \frac{a}{b}& \text{ for }a\cdot \left(\frac{1}{b}\right)\\ a^{-1}& \text{ sometimes for }\frac{1}{a}\end{array}$$

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Natural Numbers ( $\mathbb{N}$)

$$\mathbb{N}={\mathbb{Z}}^{>0}$$