MATH 87430 - Topics in Number Theory: Algebraic & Transcendental Numbers [91879]

Professor: Kevin O'Bryant

Prerequisites:
We will use elementary number theory, a small amount of abstract algebra (ring extensions, integral bases), and single-variable complex functions.

Description:
The course begins with a study of diophantine approximation: how close can a rational number be to a particular algebraic number? Surprisingly, it turns out that while fractions with small denominator do `clump' around certain real numbers, the most intense clumping never happens around algebraic numbers. This provides one way to write down transcendental numbers. We will end the semester with complete proofs of the Hermite-Lindemann-Weierstrauss and Gelfond-Schneider theorems, showing that numbers such as $\pi, e, e^{\pi}, \sqrt{2}^{\sqrt{2}}$ are transcendental.