Complex analysis is one of the most rich and amazing areas of mathematics, for the complex numbers possess geometric properties that intertwine with their analytic properties in surprising and beautiful ways. This is hardly a place to discuss the subject in detail, so I will focus on Rouché’s Theorem, with an entertaining application.

Rouché’s Theorem If is an open disc in , and and are complex-valued differentiable functions defined in some neighborhood of , and if on the boundary of , then and have the same number of zeros inside , up to multiplicity.

By “multiplicity” here we mean the following: is a zero of multiplicity 1 of if but , and a zero of multiplicity two if but , etc.

The Wikipedia article on this theorem has an interesting informal summary: “If a person were to walk a dog on a leash around and around a tree, and the length of the leash is less than the radius of the tree, then the person and the dog go around the tree an equal number of times.” A little bit of an oversimplification perhaps, but an alluring hint at the deep geometric principles at work in this theorem.

An application:

Proposition: For any , the polynomial is irreducible.

Proof: Let , and let . Choose to be sufficiently close to that .

Then, when , we have , so and satisfy the conditions of Rouché’s Theorem, and have the same number of zeros inside the circle around the origin of radius . Letting , we find that and have the same number of roots inside the circle of radius .

But has exactly one root inside this circle, so , hence must have exactly one root with .

Now, suppose that were a root of with . Then , so . So lies on a circle around of radius , as well as on the circle around the origin of radius , so . Since , has no roots on the unit circle.

Now we are done! For if over the integers, then the constant terms of and must be . But these constant terms are the product of some subset of the roots of . If is a root of , since it is the only root of absolute value less than or equal to one, it follows that must have every single other root of as a root, and the factorization must be trivial.

For anyone interested in studying this subject in greater detail, I highly recommend Functions of One Complex Variable by John B. Conway. And if you want to know more about numbers like our above, read up on Pisot numbers.

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