Field axioms (for R and C)

The field axioms for a set $\mathbb{F}$ equipped with operations $+$ and $\cdot$ assert that:

• $(\mathbb{F},+)$ is an abelian group (associativity, commutativity, additive identity $0$, additive inverses),
• multiplication is associative and commutative, with multiplicative identity $1\ne 0$,
• every nonzero element has a multiplicative inverse, and
• multiplication distributes over addition: $a(b+c)=ab+ac \quad \text{for all } a,b,c\in\mathbb{F}.$

They encode the algebraic manipulation rules used throughout analysis. In real analysis, $\mathbb{R}$ and $\mathbb{C}$ are treated as fields satisfying these axioms.