|
Size: 536
Comment: Initial commit, incomplete
|
← Revision 3 as of 2026-02-16 04:24:44 ⇥
Size: 2298
Comment: Finite fields
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 17: | Line 17: |
| The multiplication must also feature these properties: | 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'''. |
Fields
Fields are sets that are closed under addition and multiplication.
Contents
Description
A field is a 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 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 Zp or Fp 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.