Field

Last updated Nov 1, 2022

Suppose $X$ is a Nonempty Set, and $+: X \times X \to X$, $*: X \times X \to X$ are two Operations on $X$. Then $(X, +, *)$ is a Field if it satisfies the following properties:

1. Additive Identity: There exists $0 \in X$ so that for all $x \in X$, $0 + x = x$.
2. Associativity of $+$: For all $x,y,z \in X$, $(x + y) + z = x + (y + z)$
3. Commutativity of $+$: For all $x, y \in X$, $x + y = y + x$
4. Additive Inverses: If $x \in X$, then there exists $-x \in X$ so that $x + (-x) = 0$
5. Multiplicative Identity: There exists $1 \in X$ so that for all $x \in X$, $1 * x = x$
6. Associativity of $*$: For all $x,y,z \in X$, $(x * y) * z = x * (y * z)$
7. Commutativity of $*$: For all $x, y \in X$, $x * y = y * x$
8. Multiplicative Inverses: If $x \in X$ and $x \neq 0$, then there exists $x^{-1} \in X$ so that $x * x^{-1} = 1$
9. Distributivity of $$ over $+$: For all $x, y, z \in X$, $x(y + z) = xy + x*z$

1. In other words, a Field is a Commutative Ring with Multiplicative Inverses. Thus, a Field carries with it all properties of Commutative Rings and its subclasses (like Groups).