# Definition:Field Axioms

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## Definition

The properties of a field are as follows.

For a given field $\struct {F, +, \circ}$, these statements hold true:

\((\text A 0)\) | $:$ | Closure under addition | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x + y \in F \) | ||||

\((\text A 1)\) | $:$ | Associativity of addition | \(\displaystyle \forall x, y, z \in F:\) | \(\displaystyle \paren {x + y} + z = x + \paren {y + z} \) | ||||

\((\text A 2)\) | $:$ | Commutativity of addition | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x + y = y + x \) | ||||

\((\text A 3)\) | $:$ | Identity element for addition | \(\displaystyle \exists 0_F \in F: \forall x \in F:\) | \(\displaystyle x + 0_F = x = 0_F + x \) | $0_F$ is called the zero | |||

\((\text A 4)\) | $:$ | Inverse elements for addition | \(\displaystyle \forall x: \exists x' \in F:\) | \(\displaystyle x + x' = 0_F = x' + x \) | $x'$ is called a negative element | |||

\((\text M 0)\) | $:$ | Closure under product | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x \circ y \in F \) | ||||

\((\text M 1)\) | $:$ | Associativity of product | \(\displaystyle \forall x, y, z \in F:\) | \(\displaystyle \paren {x \circ y} \circ z = x \circ \paren {y \circ z} \) | ||||

\((\text M 2)\) | $:$ | Commutativity of product | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x \circ y = y \circ x \) | ||||

\((\text M 3)\) | $:$ | Identity element for product | \(\displaystyle \exists 1_F \in F, 1_F \ne 0_F: \forall x \in F:\) | \(\displaystyle x \circ 1_F = x = 1_F \circ x \) | $1_F$ is called the unity | |||

\((\text M 4)\) | $:$ | Inverse elements for product | \(\displaystyle \forall x \in F^*: \exists x^{-1} \in F^*:\) | \(\displaystyle x \circ x^{-1} = 1_F = x^{-1} \circ x \) | ||||

\((\text D)\) | $:$ | Product is distributive over addition | \(\displaystyle \forall x, y, z \in F:\) | \(\displaystyle x \circ \paren {y + z} = \paren {x \circ y} + \paren {x \circ z} \) |

These are called the **field axioms**.

## Sources

- 1974: Robert Gilmore:
*Lie Groups, Lie Algebras and Some of their Applications*... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $3$. FIELD - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 19$: Properties of $\Z_m$ as an algebraic system