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Describe the kernel and fibers of a given group homomorphism


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.9

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. Describe the kernel and the fibers of $\varphi$ geometrically as subsets of the plane.


Solution: We have \begin{align*}\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).\end{align*}The image of $\varphi$ is $\mathbb R^{+}$. 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.


Linearity

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