Question

Let f: Cˣ to Cˣ ,where X= multiplication, and C= complex numbers, be defined as f(g)=z⁴. Prove that f is a homomorphism of multiplicative groups. Then find the kernel of f.

Answer 1

The function f(z)=z⁴ is a homomorphism because it satisfies the homomorphism property for complex numbers, and its kernel consists of the fourth roots of unity.

To prove that a function f: C× to C× defined by f(g)=z⁴ is a homomorphism of multiplicative groups, we must show that it satisfies the homomorphism property f(g1 * g2) = f(g1) * f(g2) for all g1, g2 in C×. Taking two complex numbers z1 and z2, we have f(z1 * z2) = (z1 * z2)⁴ = z1⁴ * z2⁴ = f(z1) * f(z2), thus satisfying the property. Next, to find the kernel of f, we look for all elements g in C× such that f(g)=1. The kernel is z ∈ C× . This includes all complex numbers which are fourth roots of unity.