Define by . Prove that is a homomorphism and find its image. Descibe the kernel and the fibers of geometrically as subsets of the plane.

We have . The fiber of is the circle in the complex plane having radius and centered at the origin; in particular, the kernel is the complex circle centered at the origin of radius 1.

## Comments

I believe the image of the map is the positive real numbers, since a^2 + b^2 > 0 for all nonzero complex numbers a + bi.

This is not surjective. It maps onto the positive real numbers.

Fixed. Thanks!