Fields

• A set of scalars on which two operations addition ($x$) and multiplication ($\cdot$) are defined so for each $x,y\in F$, $x+y\in F$ and $x\cdot y\in F$
• And the following hold $\forall a,b,c\in F$
  1. $a+b=b+a$ and $a\cdot b=b\cdot a$ (commutativity of addition and multiplication)
  2. $(a+b)+c=a+(b+c)$ and $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ (associativity)
  3. $\exists \text{ distinct }0,1\in F\text{ s.t. }0+a=a\text{ and }1\cdot a=a$ (identity elements)
  4. $\forall a\in F\text{ and }\forall \text{b}\ne 0,\exists c,d\in F\text{ s.t }a+c=0,b\cdot d=0$ (additive and multiplicative inverse)
  5. $a\cdot (b+c)=a\cdot b+a\cdot c$ (distributivity)
• Examples: $\mathbb{R},\mathbb{C},\mathbb{Q}$