Proving (a+b)^2=a^2+2ab+b^2 in Fields

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The discussion revolves around proving the equation (a+b)² = a² + 2ab + b² in any field. Participants highlight the need to define "2" as 1 + 1, emphasizing that while fields contain the number 1, they may not include all natural numbers. The concept of 2 being equal to 0 in certain fields, like F2, is acknowledged, yet it is noted that the original equation remains valid regardless. The importance of understanding field axioms, such as commutativity and associativity, is also discussed. Ultimately, the proof hinges on the definitions and properties inherent to fields.
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Homework Statement



I need to prove that in any field :

(a+b)^2=a^2+2ab+b^2

Homework Equations





The Attempt at a Solution


i don't know how to start the proof ... I know all the attributes of fields but i got stuck
 
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welcome to pf!

hi fastidious1! welcome to pf! :smile:
fastidious1 said:
... I know all the attributes of fields but i got stuck

commutative, associative, … ? :wink:
 
ok, and what about a*a=a^2 ?
do I need to supply a proof for this product?
and a*b+a*b=2ab
can i say that it is axioms?
tnx!
 
fastidious1 said:
ok, and what about a*a=a^2 ?
do I need to supply a proof for this product?

no, that's just the definition of (or another name for) a2 ! :smile:
and a*b+a*b=2ab
can i say that it is axioms?

imo, the question is a bit weird …

"2" isn't in any of the axioms

"2" needs to be defined, and the question hasn't defined it :frown:

i think you'll have to define it! (how? :wink:)​
 
what is 2 ?
2=1+1
in any field the number 1 is exist and in any field addition is already defined so we have 1+1 we call it 2. we can also can to call to all the following numbers 3'4
it is doesn't mean that any field include all the natural number. it is possible that in some condition that 2=0(like in field F2) does it correct ? tnx
 
hi fastidious1! :smile:
fastidious1 said:
what is 2 ?
2=1+1
in any field the number 1 is exist and in any field addition is already defined so we have 1+1 we call it 2. we can also can to call to all the following numbers 3'4

yup, that's perfect :smile: …

there's a 1, and 1 + 1 must be something, so we call it 2 …

of course, you still need to prove that a + a = 2a ! :wink:​
it is doesn't mean that any field include all the natural number. it is possible that in some condition that 2=0(like in field F2) does it correct ? tnx

yes …

but it doesn't matter, because the original equation would still be true! :-p
 

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