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Mathematics-Online course: Linear Algebra - Basic Structures - Groups and Fields

Fields

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A set $ K$ together with two operations (addition and multiplication) is called field if the following requirements (field axioms) are satisfied:

K1
Associative law with respect to the addition: $ (a+b)+c = a+(b+c)$.
K2
There exists an element $ 0 \in K$ so that $ a+0=0+a=a$ for all $ a \in K$ (zero element).
K3
For each $ a$ there exists an element $ b$ so that $ a+b=b+a=0$. Often $ -a$ is written instead of $ b$ (inverse with respect to the addition).
K4
Commutative law with respect to the addition: $ a+b=b+a$.
K5
Associative law with respect to the multiplication: $ (a \cdot b) \cdot c = a \cdot (b \cdot c)$.
K6
There exists an element $ 1 \in K \setminus \{0\}$ so that $ 1 \cdot a=a\cdot 1=a$ for all $ a \in K$ (identity element).
K7
For each $ a \in K \setminus \{0\}$ there is an element $ b \in K$ with $ a \cdot b=b \cdot a = 1$ (inverse with respect to the multiplication).
K8
Commutative law with respect to the multiplication: $ a \cdot b = b \cdot a$.
K9
Distributive law: $ a \cdot (b + c) = a \cdot b + a \cdot c$.

(Authors: Burkhardt/Höllig/Hörner)
The sets of rational numbers $ \mathbb{Q}$, real numbers $ \mathbb{R}$ and complex numbers $ \mathbb{C}$ are fields.

The calculation rules follow directly from the definitions of the corresponding operations. In each case the zero element is 0 and the unit element is 1 or $ 1+\mathrm{i}0$, resp.

The inverse element with respect to multiplication for a complex number $ z=x+\mathrm{i}y \neq 0$ is

$\displaystyle w=\frac{x}{x^2+y^2} +\mathrm{i}\,\frac{-y}{x^2+y^2} $

because

$\displaystyle z \cdot w =(x+\mathrm{i}y) \left(\frac{x}{x^2+y^2} +\mathrm{i}\,\... ...mathrm{i}^2y^2}{x^2+y^2}+\mathrm{i}\frac{-xy+yx}{x^2+y^2} = 1+\mathrm{i}0\,. $

  automatically generated 4/21/2005