12-240/Classnotes for Thursday September 13

From Drorbn
Revision as of 21:21, 13 September 2012 by Starash (Talk | contribs)

Jump to: navigation, search

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

\forall \!\, a, b \in \!\, F: a+b=b+a and a.b=b.a

F2: associative law

\forall \!\, a, b, c \in \!\, F: (a+b)+c=a+(b+c) and (a.b).c= a.(b.c)


F3: the existence of identity elements

\forall \!\, a \in \!\, F, a+o=a and a.1=a


F4: existence of inverses

\forall \!\, a \in \!\, F , \exists \!\, c, d \in \!\ F such that a+c=o and a.d=1


F5: contributive law

\forall \!\, a, b, c \in \!\, F, a.(b+c)=a.b + a.c

Examples

Lecture Notes, upload by Starash