12-240/Classnotes for Thursday September 13: Difference between revisions
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<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F, a.(b+c)=a.b + a.c |
<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F, a.(b+c)=a.b + a.c |
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== Examples == |
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Revision as of 16:47, 13 September 2012
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In the second day of the class, the professor continues on the definition of a field.
Definition of a field
Combined with a part from the first class, we have a complete definition as follow:
A field is a set "F' with two binary operations +,x defind on it, and two special elements 0 ≠ 1 such that
F1: commutative law
[math]\displaystyle{ \forall \!\, }[/math] a, b [math]\displaystyle{ \in \!\, }[/math] F: a+b=b+a and a.b=b.a
F2: associative law
[math]\displaystyle{ \forall \!\, }[/math] a, b, c [math]\displaystyle{ \in \!\, }[/math] F: (a+b)+c=a+(b+c) and (a.b).c= a.(b.c)
F3: the existence of identity elements
[math]\displaystyle{ \forall \!\, }[/math] a [math]\displaystyle{ \in \!\, }[/math] F, a+o=a and a.1=a
F4: existence of inverses
[math]\displaystyle{ \forall \!\, }[/math] a [math]\displaystyle{ \in \!\, }[/math] F ,[math]\displaystyle{ \exists \!\, c, d \in \!\ }[/math] F such that a+c=o and a.d=1
F5: contributive law
[math]\displaystyle{ \forall \!\, }[/math] a, b, c [math]\displaystyle{ \in \!\, }[/math] F, a.(b+c)=a.b + a.c
