Proving an Axiom: Stuck on Field's Commutativity

[sara_87]

Homework Statement

I was asked to prove the axioms of a field.
so, if we look at the first one:
commutativity: a+b=b+a and a*b=b*a where a and b belong in the set of the field

Homework Equations

The Attempt at a Solution

it's tempting to just substitute values but i know this is the wrong approach since this wouldn't prove the axiom for all values in the set. so I am stuck.

 

[aPhilosopher]

It would help a lot if we knew what the set was.

 

[sara_87]

the question just says: prove the axioms of a field.
but the chapter we are studying is complex numbers so maybe i have to prove them for complex numbers. if so, would this proof be good enough:
for: a+b=b+a: where a=u+vi and b=x+yi
u+vi+x+yi=x+yi+u+vi
??

 

[aPhilosopher]

mmmmm. I suppose you can't just say that C is isomorphic to R[x]/(x² + 1) and that because (x² + 1) is maximal, it must be a field?

How you write the proof depends on how you've defined complex numbers and how you've defined addition and multiplication. Is the definition something like a + bi = (a, b)?

 

[sara_87]

what do you mean '(x^2+1) i maximal' here?
yes that's the definition we use for complex numbers.

 

[aPhilosopher]

Ok, then what you want to do for commutativity of addition is (a, b) + (c, d) = (a + c, b + d) = (c + a, d + b) = (c, d) + (a, b).

Do you notice how we "got a free ride" once we got a and c in the same tuple? That just reduces to commutativity of the reals, which I assume you can take for granted.

As far as the maximal thing, it just means that the ideal generated by x¹ + 1 in the polynomial ring over R isn't contained in any others. You'll probably learn about it if you're taking an algebra class. It was mainly just a joke.

 

[sara_87]

:) ok
I understand the proof now.
for a*b=b*a, would this be convincing:
(a,b)*(c,d)=(a+bi)(c+di)=(ac-bd, bc+ad)=(ca-db, cb+da)=(c,d)*(a,b)

 

[aPhilosopher]

(a,b)*(c,d)=[STRIKE](a+bi)(c+di)[/STRIKE]=(ac-bd, bc+ad)=(ca-db, cb+da)=(c,d)*(a,b)

You don't need to stick the term that I struck out in there. Remember, i is just (0, 1). Other than that, it looks great though.

 

[sara_87]

thank you. now i can prove the other axioms.