Endomorphisms of the Complex Field

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

An endomorphism phi from the complex numbers to itself is a map that preserves the field structure, meaning it preserves addition and multiplication.

We are specifically looking for endomorphisms that preserve the real numbers. This means for any real number r, the image of r under the function is equal to r itself.

Because the complex numbers can be expressed as a plus b i, where a and b are real numbers, the behavior of the function on any complex number is determined by its value on the imaginary unit i.

Now, let's look at the property of the imaginary unit. Since i squared equals negative one, applying the function to both sides gives us f of i squared equals f of negative one.

Since f is a field homomorphism, f of i squared is the same as the square of f of i, and f of negative one must be negative one because it is a real number.

This gives us a simple quadratic equation for f of i. The square of f of i must be negative one.

In the complex field, the only values whose square is negative one are i and negative i.

Let's explore the first case, where f of i is equal to i.