Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

math_proof

  • 2 years ago

field proof A field consists of a set F, distinguished elements 0 ∈ F, 1 ∈ F, and functions + : F ×F → F , and · : F × F → F . We will write a + b for +(a, b) and ab for ·(a, b). These data are subject to the axioms: i) ∀a,b,c∈F, (a+b)+c=a+(b+c) ii) ∀a∈F, 0+a=a iii) ∀a∈F, ∃a ̃∈F, a ̃+a=0 iv) ∀a,b∈F, a+b=b+a v) ∀a,b,c ∈ F, (ab)c = a(bc) vi) ∀a∈F, 1a=a vii) ∀a∈F, ∼(a=0)⇒∃a′ ∈F,a′a=1 viii) ∀a, b ∈ F, ab = ba ix) ∀a,b,c∈F, (a+b)c=ac+bc

  • This Question is Closed
  1. math_proof
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Question 2 Prove the following results about fields; use only the axioms or a fact that you have already proved. Do not jump to conclusions based on what happens with the real numbers; note that division and subtraction have not been defined, and that you cannot divide by 1+1 since it could be zero. Each line of your argument should specify which axioms or previous results are used. (1) If a+a=a then a=0. (2) ∀a∈F, 0a=0 (3) If 0 = 1, then ∀a ∈ F,a = 0. Hence we usually assume that 0 ̸= 1; (this is sometimes rephrased as “F has at least two elements”). (4) Ifx∈F is an element such that∀a∈F,x+a=a then x=0 (5) Ifa∈F andx∈F is an element such that x+a=0,then x=a' (6) ∀a∈A,a''=a (7) ∀a∈1'a=a'

  2. jarmvel
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Is nor very complicated. For example for the first (1) a+a=a then a=0 proof: \[a \in F \Rightarrow \exists -a \in F\] then we have that, using i), ii) and iii) \[(a+a)+(-a)=a+(a+(-a))=a+0=a=a+(-a)=0\]

  3. math_proof
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @jarmvel but the subtraction has not been defined

  4. jarmvel
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Oh! sorry. I'm not using subtraction. I'm denoting the inverse of \[a\] by \[-a\]

  5. jarmvel
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    additive inverse, of course. :)

  6. math_proof
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ooo alright

  7. math_proof
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is there special notation of additive inverse?

  8. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.