Teaching materials for the course "Topics in number theory"

S.V.Duzhin,

Lecture 1

Paper by Nathaniel Johnston about the derivation of Conway's polynomial.

Book edited by Thomas, Cover and Gobinath and containing Conway's original article on the subject (pp. 179--194).

Table of transitions between Conway's elementary blocks.

Maple program to compute Conway's polynomial, together with the result of its work.

Lecture 2

Barry Mazur. Algebraic numbers. (A review of basic ideas.)

S.Lang. Algebra. (Proof that the field of algebraic numbers is algebraically closed).

S.Lang. Algebraic numbers. (Rather advanced book.)

van der Waerden. Algebra. (Proof that the field of algebraic numbers is algebraically closed).

V.Prasolov. Polynomials. (Resultants, Tchirnhaus transformation, criteria of irreducibility.)

The sum of two algebraic numbers: Dr.Jacques. (Elementary.)

I.M.Isaacs. Degrees of sums in separable field extensions. (Proved that if deg(a)=m, deg(b)=n and m,n are mutually prime then deg(a+b)=mn).

Drungilas et al. On the sum of two algebraic numbers.

Anonymous. The Sylvester resultant. (On the resultant of a pair of bivariate polynomials).

Maple program to find the degree of 3^(1/2)+s^(1/3) and the result of its work.

Lectures 3-4

Paper by Churchhouse and Muir

Paper by Stark.