Define $\phi : \mathbb{C}^x \mapsto \mathbb{R}^x$ by $\phi(a+bi) = a^2 + b^2$. Prove that $\phi$ is a homomorphism and find the image of $\phi$. Describe the kernel and the fibers of $\phi$ geometrically (as subsets of the plane).
Part One
Proving that it is a homomorphism is trivial and I did it correctly. I was hoping to receive a review of the other three parts.
The image of $\phi$ is the set of all positive real numbers. (not sure how to "prove" this as it seems rather trivial. Should I maybe do a proof by contradiction, or is it trivial enough to state without proof?)
Part Two
We define the kernel of $\phi$ as
$\{g \in G | \phi(g) = 1\}$
If $\phi(g) = 1$, then $1 = a^2 +b^2$. Obviously, this means either $a^2 =1$ or $b^2=1$, i.e. $a=1, a=-1, b=1,$ or $b=-1$. Note that this implies that the kernel of $\phi = \{\pm1,\pm i\}$.
(particularly concerned about the next part)
For any value $\phi(x) = c^2$, for any $x\in \mathbb{C}^x$, $x$ can be of the form $\pm c+0i$ or $0\pm ci$. From this we can generalize, since addition is commutative that $\forall a^2+b^2 \in \mathbb{R}^x$
$X_{a^2+b^2} = \{\pm a \pm bi, \pm b \pm ai\}$
Best Answer
To show the image of $\phi$ is every positive number, just take $x=x+0i$. For the kernel, consider the points in the unit circle; remember the identity :$sin^2x+cos^2x=...$