= Fields = '''Fields''' are [[Analysis/Sets|sets]] that are closed under addition and multiplication. <> ---- == Description == A field is a [[Analysis/Sets|set]] for which a pair of binary operations are defined: addition and mutliplication. The field must be closed under both operations; that is, addition is a [[Analysis/Functions|map]] as ''F + F -> F'' and multiplication is a map as ''F × F -> F''. The addition operation must also feature these properties: * commutivity: ''a + b = b + a'' * associativity: ''(a + b) + c = a + (b + c)'' * There exists a value satisfying an identity property: ''a + 0 = a''. * invertibility: ''a + (-a) = 0'' The multiplication operation must also feature these properties: * commutivity: ''ab = ba'' * associativity: ''(ab)c = a(bc)'' * There exists a value satisfying an identity property: ''a1 = a''. * invertibility: Formally, because zero is excluded, it is said that ''∀ a ∈ F \ {0} aa^-1^ = 1''. Lastly there js a distributivity property: ''(a + b)c = ac + bc''. ---- == Finite Fields == A non-obvious example of a field is the set ''{0,1}'' for which addition is defined ''(a + b) % 2'' and for which multiplication is defined ''(ab) % 2''. Because of the modulus, every possible arithmatic operation is: * ''0 + 0 = 0'' * ''0 + 1 = 1'' * ''1 + 1 = 0'' * ''0 × 0 = 0'' * ''0 × 1 = 0'' * ''1 × 1 = 1'' The invertibility property of multiplication is the trickiest to satisfy, but due to the modulus, the inverse of 1 is in fact 0. (Recall that 0 is not required to have an inverse.) Consider then the set ''{0,1,2}'' where addition is defined ''(a + b) % 3'' and for which multiplication is defined ''(ab) % 3''. It follows that ''1 × 1 = 1'' and ''2 × 2 = 1'', satisfying the invertibility property of multiplication again. A '''finite field''' can be defined by the integers with any prime modulus. They are usually notated ''Z,,p,,'' or ''F,,p,,'' where ''p'' is the modulus. ---- == Commutative Rings == There are many sets which satisfy some but not all properties of a field. For example, the set of integers does not contain multiplicative inverses. It is instead a '''commutative ring'''. ---- CategoryRicottone