Fields

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Field”,“label”:“field”,“_index”:0}] Definition 1 (Field). A field is a set $F$ together with 2 binary operations(addition and multiplication) that satisfy the below axioms.

1. Associativity
2. Commutativity
3. Additive Inverses
4. Additive Identity
5. Multiplicative Identity

Axioms §

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Associativity”,“label”:“associativity”,“_index”:1}] Axiom 2 (Associativity). $(a+b)+c=a+(b+c)$ and $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ for $a,b,c\in F$

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Commutativity”,“label”:“commutativity”,“_index”:2}] Axiom 3 (Commutativity). $a+b=b+a$ and $a\cdot b=b\cdot a$.

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Identities”,“label”:“identities”,“_index”:3}] Axiom 4 (Identities). $\exists$ two distinct elements $0,1$ s.t. $a+0=a$ and $a\cdot 1=a$ for $a\in F$.

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Inverses”,“label”:“inverses”,“_index”:4}] Axiom 5 (Inverses). For $a\in F$,

1. $\exists$ $-a$ s.t. $a+(-a)=0$ and
2. $\exists$ $a^{-1}$ s.t. $a\cdot a^{-1}=1$.

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Distributivitiy”,“label”:“distributivitiy”,“_index”:5}] Axiom 6 (Distributivitiy). $a\cdot (b+c)=ab+ac$.

Examples §

• Fields: $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$
• Non-fields: $\mathbb{Z}$ with addition and multiplication

Binary Field §

• unique field structure on the set $F=\{0,1\}$
• example of a finite field
• suffices to determine addition and multiplication tables
• addition table:

	0	1
0	0	1
1	1	0

• $1+1$ must equal $0$ since $1+1=1⟹1=0$ which violates ^18bb35
• multiplication table:

	0	1
0	0	0
1	0	1