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

[fastidious1]

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

 

[tiny-tim]

welcome to pf!

hi fastidious1! welcome to pf!

... I know all the attributes of fields but i got stuck

commutative, associative, … ?

 

[fastidious1]

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!

 

[tiny-tim]

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 !

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

i think you'll have to define it! (how? )​

 

[fastidious1]

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

 

[tiny-tim]

hi fastidious1!

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 …

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 ! ​

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!