Field

==2024-12-02

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Theory

A field is a Mathematical Triple $(F,+,\cdot )$ where $F$ is a set, and $+$ and $\cdot$ are binary operations on $F$ (called addition and multiplication respectively) satisfying the following nine conditions.

These conditions are called The Field Axioms.

Considerations: $a,b,c\in F$

1. (Commutativity of Addition): $a+b=b+a$

2. (Associativity of addition.) Addition $(+)$ is an associative operation on $F$. i.e $(a+b)+c=a+(b+c)$

3. (Existence of additive identity.) There is an identity element for addition. i.e $a+0=0+a=a$, 0 being additive identity. We know that this identity is unique, and we will denote it by $0$.

4. (Existence of additive inverses.) Every element $x$ of $F$ is invertible for $+$. i.e $a+(-a)=(-a)+a=0$. We know that the additive inverse for $x$ is unique, and we will denote it by $-x$.

5. (Commutativity of multiplication.) Multiplication $(\cdot )$ is a commutative operation on $F$. i.e $a.b=b.a$

6. (Associativity of multiplication.) Multiplication is an associative operation on $F$. i.e $a.(b.c)=(a.b).c$

7. (Existence of multiplicative identity.) There is an identity element for multiplication.

We know that this identity is unique, and we will denote it by $1$.

8. (Existence of multiplicative inverses.) Every element $x$ of $F$ except possibly for $0$ is invertible for $\cdot$.

We know that the multiplicative inverse for $x$ is unique, and we will denote it by $x^{-1}$. We do not assume $0$ is not invertible. We just do not assume that it is.

9. (Distributive law.) For all $x,y,z$ in $F$, $x\cdot (y+z)=(x\cdot y)+(x\cdot z)$.

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10. (Zero-one law.) The additive identity and multiplicative identity are distinct and unique; i.e., $0\ne 1$.

We often speak of the field $F$ instead of the field $(F,+,\cdot )$.

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