12-240/Classnotes for Thursday September 13

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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

a, b F: a+b=b+a and a.b=b.a

F2: associative law

a, b, c F: (a+b)+c=a+(b+c) and (a.b).c= a.(b.c)


F3: the existence of identity elements

a F, a+o=a and a.1=a


F4: existence of inverses

a F , c, d F such that a+c=o and a.d=1


F5: contributive law

a, b, c F, a.(b+c)=a.b + a.c

Significance

Identity uniqueness It makes sense to define an operation -: F -> F called "negation"

For a F define -a to be equal that b F for which a+b=0, i.e, a+(-a)=0

Ex: F(5)={0,1,2,3,4}, define +,x

Question: What is(-3)?

Answer: -3=2

Lecture Notes, upload by Starash