12240/Classnotes for Thursday September 13

In the second day of the class, the professor continues on the definition of a field.
Contents 
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 , a+0=a and a.1=a
F4: existence of inverses
a F \0, 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
Theorems
Theorem 1: Cancellation laws a, b, c F
if a+c=b+c, then a=b
if a.c=b.c and c0, then a=b
Theorem 2: Identity uniqueness
Identity elements 0 and 1 mentioned in F3 are unique
a, b, b' F
if a+b=a and a+b'=a, then b=b'=0
if a.b=a and a.b'=a and a0, then b=b'=1
Theorem 3: Inverse uniqueness
Elements c and d mentioned in F4 are unique
a, b, b' F
if a+b=0 and a+b'=0, then b=b'
if a.b=1 and a.b'=1, then b=b'
Theorem 4
a F
( a) = a
Theorem 4
a F
0.a= 0
Significance
inverse 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 1: What is(3)?
Answer: 3=2 and 3 is unique
Similarly, the inverse uniqueness also makes sense a^(1)
identity uniqueness