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.

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  • tjm1991  On January 12, 2014 at 5:02 pm

    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.

  • 032-234  On October 5, 2014 at 11:19 am

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

    • nbloomf  On October 5, 2014 at 10:55 pm

      Fixed. Thanks!

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