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November 11th, 2016, 06:35 AM   #1
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Fields

I'm learning about fields and I still dont get how to prove things in fields, for example
1. F∈a.
and I need to prove that if a^2 = 1 , then a = 1 or a= -1
2. if a^2 =a , then a =0 or a = 1

This what I did :
1. a^2 =1 --> a^2 -1 = 0 --> a^2 -1 *1=0 (1 is neutral -axiom)-->
a^2 -1^2 --> (a+1)(a-1) = 0 --> a=1 or a=-1 .

2. a^2 - a = 0 --> a*(a-1) = 0 --> a=0 or a= 1.

is that good or should I use more fields axioms ?
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November 22nd, 2016, 05:56 PM   #2
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Quote:
Originally Posted by dave22 View Post
I'm learning about fields and I still dont get how to prove things in fields, for example
1. F∈a.
You mean a∈ F.

Quote:
and I need to prove that if a^2 = 1 , then a = 1 or a= -1
2. if a^2 =a , then a =0 or a = 1
This what I did :
1. a^2 =1 --> a^2 -1 = 0 --> a^2 -1 *1=0 (1 is neutral -axiom)-->
a^2 -1^2 --> (a+1)(a-1) = 0 --> a=1 or a=-1 .
Before the last statement, say first that a+ 1= 0 and a- 1= 0. And that is true because F, being a field, does not have "zero divisors".

Quote:
2. a^2 - a = 0 --> a*(a-1) = 0 --> a=0 or a= 1.
Again, first say that either a= 0 or a- 1= 0.

Quote:
is that good or should I use more fields axioms ?
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