Proving that a subspace of C is a field

[lolittaFarhat]

Homework Statement

Show that the only m for which the subspace of C given by {z ∈ C: Im(z) = m Re(z)} is a field is m=0.

Homework Equations

Field axioms

The Attempt at a Solution

I tried to prove one direction :
- If z is in the subspace, Re z>0 and m≠0 then Arg z<Arg z^2, so z^2 is not in the subspace.
I need a hint to prove the other direction.

 

[Mark44]

Homework Statement

Show that the only m for which the subspace of C given by {z ∈ C: Im(z) = m Re(z)} is a field is m=0.

Homework Equations

Field axioms

The Attempt at a Solution

I tried to prove one direction :
- If z is in the subspace, Re z>0

No. For example, z = -1 -mi is in C, but Re(z) < 0. Note that for a given value of m, C is a straight line through the origin. Which field axioms prevent this subspace from being a field?

and m≠0 then Arg z<Arg z^2, so z^2 is not in the subspace.
I need a hint to prove the other direction.

 

[RUber]

An element of the subspace z would have the form z = x + imx, with x in the reals.
As Mark mentioned, you should check the axioms, it seems like this space satisfies most of the addition axioms. I think you were on the right track with squaring z. Using z^2 as an example, you can show that the only m that satisfies multiplication axioms (for any choice of x and a fixed m) is 0.