Fundamental Theorem of Algebra

The fundamental theorem of algebra states that every nonconstant polynomial with complex coefficients has a complex root. In fact, every known proof of this theorem involves a bit of analysis, since a purely algebraic construction of the complex numbers is very hard to work with.

One of the shortest proofs of the fundamental theorem of algebra uses Liouville's Theorem of complex analysis, which says that a bounded entire function is constant. Suppose that $p(x)$ were a complex polynomial with no complex roots. Then $1/p(x)$ would be an entire function. To see that it is bounded, we must show that $|p(x)|\ge c>0$ for some $c$ . It can easily be shown that outside a sufficiently large disc, $p(x)$ is dominated by its leading term, which gets large in absolute value, so it suffices to work inside a disc of radius $R$ for some $R$ to find the minimum value. But the (closed) disc of radius $R$ is a compact set, so $|p(x)|$ must achieve a minimum inside the disc. This minimum value is $c$ . Hence $\left|\frac{1}{p(x)}\right|\le\frac{1}{c}$ , so $1/p(x)$ would be a bounded entire function and hence constant. This proves the theorem.

(This ought to be cleaned up.)

There is another proof that uses group theory and Galois theory that is very nice, and someone (possibly me) should write it here at some point. The advantage of this proof is that it uses only a very small amount of analysis (just the Intermediate Value Theorem), and the rest is all actually algebra.

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Fundamental Theorem of Algebra