Endomorphisms of the complex field

Hi nguyệt, let's find all the endomorphisms of the complex field that preserve real numbers.

An endomorphism of the complex field is a ring homomorphism from the complex numbers to itself. We are looking for maps sigma such that sigma of r equals r for every real number r.

Since every complex number z can be written as a plus b times i, where a and b are real, we can apply the properties of the homomorphism.

Applying sigma to z, and using the fact that sigma preserves addition and multiplication, we get sigma of a plus sigma of b times sigma of i.

Because the map preserves real numbers, sigma of a is simply a, and sigma of b is simply b.

This shows that the entire map is determined by what happens to the imaginary unit i.

We know that i squared equals negative one. Applying the homomorphism to both sides, we get sigma of i squared equals sigma of negative one.