1 Field
We are used to dealing with the operations addition and multiplication in sets such as the real numbers \(\mathbb{R}\) or the complex numbers \(\mathbb{C}\) which we are told follow certain rules such as commutativity and associativity. Generally, these are examples of fields \(F\) and the rules that addition and multiplication follow are field axioms.
A field \(F\) is a set of elements upon which is defined operations of addition and subtraction that satisfy the field axioms.
The field axioms are defined as follows:
- \(x,y \in F \implies x + y \in F\) (addition is a binary operation);
- \(x,y \in F \implies xy \in F\) (multiplication is a binary operation);
- \(x,y \in F \implies x + y = y + x\) (addition is commutative);
- \(x,y \in F \implies xy = yx\) (multiplication is commutative);
- \(x,y,z \in F\implies (x+y)+z=x+(y+z)\) (addition is associative);
- \(x,y,z \in F \implies (xy)z=x(yz)\) (multiplication is associative);
- \(\exists 0 \in F:\forall x \in F,~x + 0=x\) (additive identity);
- \(\exists 1 \in F:\forall x \in F,~x \times 1=x\) (multiplicative identity);
- \(\forall x \in F,~ \exists y \in F:x+y=0\) (additive inverses);
- \(\forall x \in F,~\exists y\in F:x \times y=1\) (multiplicative inverses);
- \(\forall x, y,z \in F,~x\times(y+z)=(x \times y)+(x \times z)\) (multiplication distributes over addition).
The rational numbers \(\mathbb{Q}\), real numbers \(\mathbb{R}\) and complex numbers \(\mathbb{C}\) are all examples of fields.