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Linear Algebra Fields Proof

  1. Sep 17, 2013 #1
    1. The problem statement, all variables and given/known data

    Let K be the closure of Qu{i}, that is, K is the set of all numbers that can be obtained by (repeatedly)
    adding and multiplying rational numbers and i, where i is the complex square root of 1.
    Show that K is a Field.

    2. Relevant equations



    3. The attempt at a solution
    I am having trouble starting on this problem:

    What I know:

    Proof the Zero vector is in the set
    Proof both addition and scalar multiplication
    proof additive and multiplicative inverse

    ^ am I missing anything?

    And i am guessing I have to prove it in the form of
    let Q be rational numbers
    and scalars a and b in F (field)

    aQ + bi = K
     
  2. jcsd
  3. Sep 17, 2013 #2

    Mark44

    Staff: Mentor

    I'm sure you mean √(-1).
    Well, yes, quite a lot.
    For starters, you're not dealing with vectors. Your textbook should have a definition of the axioms that define a field. You can also find them here, in the section titled "Definition and illustration" - http://en.wikipedia.org/wiki/Field_axioms.

    What "scalars" are you talking about? You need to show that a particular set, together with the operations of addition and multiplication, satisfy all of the field axioms.
     
  4. Sep 17, 2013 #3
    how do I start proofing this? I dont think i have to proof an entire list of axioms?
     
  5. Sep 17, 2013 #4

    Mark44

    Staff: Mentor

    You prove (not proof) that a set K and two operations constitute a field by showing that all of the axioms are satisfied. Again, the axioms should be listed in your book, and are also listed in the link I posted.

    You can start by listing a couple of arbitrary members of the set.
     
  6. Sep 18, 2013 #5
    can you give me an example? of a member of the set?
     
  7. Sep 18, 2013 #6

    Mark44

    Staff: Mentor

    2/3 + (5/6)i
     
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