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

In summary, the homework statement is that in any field, (a+b)^2=a^2+2ab+b^2. 2 must be defined for the equation to be true. a+a is not equal to 2 in any field, however the original equation would still be true.
  • #1
fastidious1
3
0

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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  • #2
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:
 
  • #3
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!
 
  • #4
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:)​
 
  • #5
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
 
  • #6
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! :tongue2:
 

1. What is "proof of attribute" in fields?

Proof of attribute in fields is a mathematical concept used in cryptography. It refers to the ability to prove that a certain attribute or property holds for a specific element in a mathematical field. This is important in the field of cryptography as it allows for verification of certain properties without revealing any sensitive information.

2. How is "proof of attribute" different from "proof of knowledge"?

Proof of attribute is different from proof of knowledge in that it focuses on proving a specific attribute or property of an element, whereas proof of knowledge involves proving that a person possesses certain knowledge or information without revealing the knowledge itself.

3. What is the significance of "proof of attribute" in cryptography?

Proof of attribute is significant in cryptography as it allows for secure verification of attributes without revealing sensitive information. This is important in applications such as digital signatures, where the signer wants to prove their identity without disclosing their private key.

4. What are some applications of "proof of attribute" in fields?

Proof of attribute in fields has various applications in cryptography, including digital signatures, secure authentication, and anonymous credentials. It is also used in more advanced cryptographic protocols such as zero-knowledge proofs and secure multi-party computation.

5. How is "proof of attribute" implemented in practice?

There are various methods for implementing proof of attribute in fields, including bilinear pairings and zero-knowledge proofs. These methods involve complex mathematical algorithms and techniques to securely prove the desired attribute without revealing any sensitive information. Implementation also depends on the specific cryptographic protocol being used.

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