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Linear Algebra Fields Proof

  • Thread starter 1LastTry
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  • #1
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Homework Statement



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.

Homework Equations





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
 

Answers and Replies

  • #2
33,270
4,967

Homework Statement



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.
I'm sure you mean √(-1).
Show that K is a Field.

Homework Equations





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?
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.

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
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.
 
  • #3
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how do I start proofing this? I dont think i have to proof an entire list of axioms?
 
  • #4
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4,967
how do I start proofing this? I dont think i have to proof an entire list of axioms?
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.
 
  • #5
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can you give me an example? of a member of the set?
 
  • #6
33,270
4,967
2/3 + (5/6)i
 
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