Field

warning

A field in mathematics does not have the same definition as a field in physics!

You Learnt This!

In primary school, you would've been taught the four basic operations: addition, subtraction, multiplication, and division $(+, -, \times, \div)$.

Subracting a number is the same as adding the negative of that number.

Example

$$10-5=10 + (-5) = 5$$

General Case

$$a-b= a + (-b)$$

Division is the same as multiplication by the reciprocal.

Example

$$25 \div 5 = 25 \times \left( \frac{1}{5} \right) = \frac{25}{5} = 5$$

General Case

$$a \div b = a \times \left( \frac{1}{b} \right) = \frac{a}{b}$$

Hence, we can say that there are only two elementary operations $(+, \times)$.

tip

You will want to get into the habit of using a dot for multiplication instead of a cross, meaning that you should write $(+,\, \cdot \;)$ instead of $(+, \times)$. This is to avoid confusion with the cross product!

The Formal Definition

With that out of the way, let's formally define what a field is.

note

You may find that the formal definition of a field can differ slightly between authors; however, they all have the same meaning.

A field $\mathbb{F}$ is a set with two operations:

• Addition

$$+:\mathbb{F}\times\mathbb{F}\rightarrow\mathbb{F}$$

• Multiplication

$$\cdot :\mathbb{F}\times\mathbb{F}\rightarrow\mathbb{F}$$

These operations must satisfy the following axioms:

• Associativity (Additive)

$$\forall a, b, c \in \mathbb{F},\quad a + (b + c) = (a + b) + c$$

• Associativity (Multiplicative)

$$\forall a, b, c \in \mathbb{F},\quad a \cdot (b \cdot c) = (a \cdot b) \cdot c$$

• Commutativity (Additive)

$$\forall a, b \in \mathbb{F},\quad a + b = b + a$$

• Commutativity (Multiplicative)

$$\forall a, b \in \mathbb{F},\quad a \cdot b = b \cdot a$$

• Additive Identity

$$\forall a \in \mathbb{F}, \exists \, e_a:\quad a + e_a = e_a + a = a$$

• Multiplicative Identity

$$\forall a \in \mathbb{F}, \exists \, e_m:\quad a \cdot e_m = e_m \cdot a = a$$

• Additive Inverse

$$\forall a \in \mathbb{F}, \exists (-a):\quad a + (-a) = e_a$$

• Multiplicative Inverse

$$\forall a \in \mathbb{F}, \exists \, a^{-1}\quad a \cdot a^{-1} = e_m$$

• Distributivity of Multiplication over Addition

$$\forall a, b, c \in \mathbb{F},\quad a \cdot (b + c) = (a \cdot b) + (a \cdot c)$$

• Distinct Additive and Multiplicative Identities

$$e_m \neq e_a$$

The standard real number line, as you know it, is a field: $\mathbb{F_{\mathbb{R}}}=(\mathbb{R}, +, \cdot)$.