Field

A field is a set on which:

addition (and its inverse of subtraction)
multiplication (and its inverse of division)
Are all defined and behave as the corresponding operations on rational (and consequently real) numbers. The operations above must satisfy:
Associativity: $a + (b + c) = (a + b) + c$ and $a \cdot (b c) = (a b) \cdot c$
Commutativity: $a + b = b + a$ and $a b = b a$
Additive Identity: $\exists 0$ where $a + 0 = a$
Multiplicative identity: $\exists 1$ where $a \cdot 1 = a$
Additive inverse: $\exists - a$ where $a + (- a) = 0$
Multiplicative inverse: $\exists a^{- 1}$ such that $a \cdot a^{- 1} = 1$
Distributivity: $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$

We note that $Q$ is a field, where:

Commutative
Associative
Left distributive $a (b + c) = a b + a c$
Additive identity $1 / a$
Additive inverse $- a$
Multiplicative Identity $1$
Multiplicative Inverse $0$
are the properties that must hold. Note that $N, Z$ are not fields but ${0, 1, 2, 3, 4}$ is a field considering the addition and multiplication operators are done $mod 5$ .

Additive Identity

\neq

Multiplicative identity

Only other than the trivial field ${0}$ , all other fields don't have their additive identity equal to their multiplicative identity. See https://en.wikipedia.org/wiki/Additive_identity

In the Context of Groups

When talking about Groups we can redefine what properties must hold for a field:

field (groups)

A field is a set $F$ together with two binary operations $+$ and $\cdot$ such that $(F, +)$ is an Abelian Group (call its identity $0$ ) and $(F - {0}, \cdot)$ is also an abelian group, and the following distributive law holds:

a \cdot (b + c) = (a \cdot b) + (a \cdot c) \forall a, b, c \in F