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Complexity theory is a mathematical study of algorithmic problems with regard to how cost-effective solutions they have. Most prominent are the decision problems, which are yes-no questions for which answer depends on some further input, for example, "Is $latex n$ a prime number?" is a decision problem depending on the input $latex n$. I assume that …

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Starting a new project – Solutions to exercises in Marcus’ book (+some info)

The book "Number Fields" by D. Marcus is a very well-known introductory book on algebraic number theory. Its most memorable aspect is, without a doubt, the great number of exercises it contains. They vary from short(ish) computational exercises, through various technical results used later in the book, to series of exercises aimed to establish (sometimes …

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Number of irreducible polynomials over a finite field

Warning to PROMYS students: this blog post contains major spoilers regarding problems 4 and 6 of the Open Door Problem Set. Read at your own risk. The following problem has appeared as problem P3 on the short exam #2 during PROMYS Europe 2016: Let $latex p,n\in\mathbb N$ with $latex p$ prime. How many irreducible polynomials …

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Discriminant and different

Discriminant of a number field is arguably its most important numerical invariant. Quite closely connected to it is the different ideal. Here we discuss the most basic properties of these two concepts. The discussion mostly follows this expository paper by K. Conrad, but also takes from a series of exercises in chapter 3 of Number Fields. Discriminant …

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Proof of the Riemann hypothesis… for polynomials

This blog post consists of three parts. The first of them contains a somewhat nontechnical description of Riemann hypothesis. In the second one we discuss what the "correct" analogue of Riemann hypothesis is for polynomials over a finite field. Finally, in the last section, we prove the Riemann hypothesis for polynomials. Riemann zeta function and …

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