Chebotarev Density Theorem – its history and applications

When/Where:

April 15, 2014, at 1:55pm, in LIT 368

Abstract:

The Chebotarev Density Theorem states that the frequency of the occurrence of a given splitting pattern of prime ideals, for all primes \(p\) less than a large integer \(N\), tends to a certain limit as \(N\) goes to infinity. It generalizes Dirichlet’s theorem on arithmetic progressions and the Frobenius Density Theorem. The work of Chebotarev led the way to the Artin Reciprocity Theorem. The theorem yields many important applications including the reduction of the problem of classifying Galois extensions of a number field to that of describing the splitting of primes in extensions.

We will discover the history of the theorem from Gauss Quadratic Reciprocity to Artin Reciprocity Law. We will learn of the struggles and brilliance of Nikolai Chebotarev. Finally, we will catch a glimpse of Chebotarev’s counterintuitive proof to his density theorem.

“I belong to the old generation of Soviet scientists, who were shaped by the circumstance of a civil war. I devised my best result while carrying water from the lower part of town to the higher part, or buckets of cabbages to the market, which my mother sold to feed the entire family.” – Nikolai Chebotarev