Fundamental Theorem of Algebra

In this post, I want to talk about the Fundamental Theorem of Algebra. Really, this could be called “Why the Complex Numbers are Cool.” For a while, I don’t really think I appreciated what this theorem was saying, but now I think I have a better way to think about it. Let’s start by stating the Fundamental Theorem of Algebra.

Theorem: Every polynomial with coefficients in \mathbb{C} of degree n has exactly n roots in \mathbb{C}.

Let’s look at an example to see this in action.

Example: Consider the polynomial f(x) = x^3+x^2+x+1. This polynomial equals zero in three places (when x=i, x=-i,x=-1). This definitely satisfies the theorem because f(x) has degree 3. (Remember that the degree of a polynomial refers to the power of the highest degree term.)

To get an appreciation for why this theorem is REALLY about the complex numbers as a field, let’s look at an example of a polynomial that has coefficients in a different field.

We’ll do an example using the field \mathbb{Z}_3. First, we can note that \mathbb{Z}_3 is a ring because we can define addition and multiplication on it and the set is closed under both of these operations. Also, it has an additive identity and multiplicative identity and every element has an additive inverse. This ring is also a field because every non zero element has a multiplicative inverse that is also in the set. We can see this since 1^2 = 1 and 2^2 = 4 mod 3 \equiv 1. So, in \mathbb{Z}_3, every nonzero element is its own inverse! Kind of cool! Now, let’s look at a polynomial with coefficients in \mathbb{Z}_3

Example: Consider the polynomial g(x)=x^2+1, where we consider the coefficients of g(x) as being elements of \mathbb{Z}_3. Now, in searching for zeros of this function, we’re restricted to looking at values that are in \mathbb{Z}_3. This isn’t that hard to do since there are only three values to check.

g(0) = (0^2+1) mod 3 \equiv 1.
g(1) = (1^2+1) mod 3 \equiv 2.
g(2) = (2^2+1) mod 3 \equiv 2.

We take each output mod 3 because we are in the field \mathbb{Z}_3.

So after building a polynomial with coefficients in \mathbb{Z}_3, it has no zeros in \mathbb{Z}_3. This shows that the complex numbers are really special as far as fields go! In fact, here’s another way that the Fundamental Theorem of Algebra is sometimes stated.

Theorem: The field of complex numbers is algebraically closed.

We can’t say this about \mathbb{Z}_3 since we built a polynomial with coefficients from that field that doesn’t have zeros in that field.

The last thing I want to say about the Fundamental Theorem of Algebra is that it’s an existence proof! Existence proofs show up all over different areas of math (analysis, number theory,…other places (?)). Why is this an existence proof? Because for any degree n polynomial with coefficients in \mathbb{C}, the theorem claims that THERE EXIST n values in \mathbb{C} that make this polynomial equal zero.

I hope this helped you care a little bit more about the Fundamental Theorem of Algebra and that you love the complex numbers just a little bit more than you did before (who thought that was even possible? Not me, for sure!).


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