## Describe the kernel and fibers of a given group homomorphism

Define $\varphi : \mathbb{C}^\times \rightarrow \mathbb{R}^\times$ by $a+bi \mapsto a^2 + b^2$. Prove that $\varphi$ is a homomorphism and find its image. Descibe the kernel and the fibers of $\varphi$ geometrically as subsets of the plane.

We have $\varphi((a+bi)(c+di))$ $= \varphi((ac-bd) + (ad+bc)i)$ $= a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2$ $= (a^2 + b^2)(c^2 + d^2)$ $= \varphi(a+bi) \varphi(c+di)$. The fiber of $c$ is the circle in the complex plane having radius $\sqrt{c}$ and centered at the origin; in particular, the kernel is the complex circle centered at the origin of radius 1.