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Field

e.g., \(\mathbb{R}\)

Definition

A field is a set on which two operation '+' and '\(\cdot\)' are defined so that exists a unique element \(x+y\in F, x\cdot y\in F\). so that following conditions hold to be true (for all elements in the field)

  1. \(a+b=b+a\) (commutativity)
  2. \((a+b)+c=a+(b+c)\) (associativity)
  3. exist distinct elements \(0,1\in F\) so that \(a+0=a,1\cdot a=a\) (existence of identities)
  4. \(\forall a\in F, \exists b\in F, a+b=0\) (additive inverse)
  5. \(\forall a\in F, \exists b\in F, ab=1\) (multiplicative inverse)
  6. \(a(b+c)=ab+bc\) (distributivity)