Proving K as a Field: Closure of Qu{i}

[1LastTry]

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

 

[Mark44]

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.

 

[1LastTry]

how do I start proofing this? I don't think i have to proof an entire list of axioms?

 

[Mark44]

how do I start proofing this? I don't 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.

 

[1LastTry]

can you give me an example? of a member of the set?

 

[Mark44]

2/3 + (5/6)i