Definition:Field (Abstract Algebra)
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Definition
A field is a non-trivial division ring whose ring product is commutative.
Thus, let $\left({F, +, \times}\right)$ be an algebraic structure.
Then $\left({F, +, \times}\right)$ is a field iff:
- the algebraic structure $\left({F, +}\right)$ is an abelian group
- the algebraic structure $\left({F^*, \times}\right)$ is an abelian group where $F^* = F \setminus \left\{{0}\right\}$
- the operation $\times$ distributes over $+$.
This definition gives rise to the field axioms, as follows:
Field Axioms
The properties of a field are as follows.
For a given field $\left({F, +, \circ}\right)$, these statements hold true:
| \((A0):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall x, y \in F:\) | \(\) | \(\displaystyle x + y \in F\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Closure under addition | |
| \((A1):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall x, y, z \in F:\) | \(\) | \(\displaystyle \left({x + y}\right) + z = x + \left({y + z}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Associativity of addition | |
| \((A2):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall x, y \in F:\) | \(\) | \(\displaystyle x + y = y + x\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Commutativity of addition | |
| \((A3):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \exists 0_F \in F: \forall x \in F:\) | \(\) | \(\displaystyle x + 0_F = x = 0_F + x\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Identity element for addition: the zero | |
| \((A4):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall x \in F: \exists x' \in F:\) | \(\) | \(\displaystyle x + x' = 0_F = x' + x\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Inverse elements for addition: negative elements | |
| \((M0):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall x, y \in F:\) | \(\) | \(\displaystyle x \circ y \in F\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Closure under product | |
| \((M1):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall x, y, z \in F:\) | \(\) | \(\displaystyle \left({x \circ y}\right) \circ z = x \circ \left({y \circ z}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Associativity of product | |
| \((M2):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall x, y \in F:\) | \(\) | \(\displaystyle x \circ y = y \circ x\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Commutativity of product | |
| \((M3):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \exists 1_F \in F: 1_F \ne 0_F: \forall x \in F:\) | \(\) | \(\displaystyle x \circ 1_F = x = 1_F \circ x\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Identity element for product: the unity | |
| \((M4):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall x \in F^*: \exists x^{-1} \in F^*:\) | \(\) | \(\displaystyle x \circ x^{-1} = 1_F = x^{-1} \circ x\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Inverse elements for product | |
| \((D):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall x, y, z \in F:\) | \(\) | \(\displaystyle x \circ \left({y + z}\right) = \left({x \circ y}\right) + \left({x \circ z}\right), \left({x + y}\right) \circ z = \left({x \circ z}\right) + \left({y \circ z}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Product is distributive over addition |
These are called the field axioms.
Internationalization
Field is translated:
| In Dutch: | lichaam | (literally: body) | ||
| In French: | corps | (literally: body) | ||
| In Spanish: | cuerpo | (literally: body) |
Also see
- Results about fields can be found here.
Sources
- Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 23$
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 4.14$
- B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.3$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 19$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 55 \ (4)$
