Field (mathematical concept)

A field is a set closed under two binary operations, with some special properties by definition.

Formal definition

Let $F:|F| \ge 2$ be a set, and $+,\cdot$ two binary operations defined over $F$.

If $\forall a,b,c \in F$:

• $a+b=b+a;a \cdot b = b \cdot a$ (commutativity)
• $(a+b)+c=a+(b+c);(a \cdot b)\cdot c = a \cdot (b \cdot c)$ (associativity)
• $\exists 0_F,1_F \in F: a+0_F=a, a \cdot 1_F=a$ (identity)
• $\exists (-a) \in F: a + (-a)=0_F; \forall a \neq 0_F: \exists (a^{-1}) \in F, a\cdot (a^{-1})=1_F$ (inverse)
• $a\cdot (b+c)=a\cdot b + a\cdot c$ (distributivity)

Then $(F,+,\cdot)$ is a field.

Example:

$F$	$(F,+,\cdot )$ is field?
$\mathbb{N}$	No, $a=1 \in \mathbb{N}: \nexists (-1) \in \mathbb{N}$
$\mathbb{Z}$	No, $a=2 \in \mathbb{N}: \nexists (2^{-1}) \in \mathbb{Z}$
$\mathbb{Q}$	Yes
$\mathbb{R}$	Yes

Alternatively, a field can be defined as a ring with additional properties:

• $a \cdot b = b \cdot a$ (multiplicative commutativity)
• $\exists 1_F \in F: a \cdot 1_F=a$ (multiplicative identity)
• $\forall a \neq 0_F: \exists (a^{-1}) \in F, a\cdot (a^{-1})=1_F$ (multiplicative inverse)

Also, note unlike for rings, there is no need to specify distributivity for each slot, since the slots for multiplication are symmetric in a field because of multiplicative commutativity.