Mathematical Topics

The topics mentioned below appear throughout the problem sets, with different topic-threads often appearing on the same set.

Euclid's algorithm

Greatest common divisor. Diophantine equation . Proof of unique factorization in .

Modular arithmetic

Inverses. Solving congruences. Fermat's Theorem. Chinese Remainder Theorem. Solving congruences .

Binomial coefficients

Pascal's triangle. Binomial Theorem. Arithmetic properties of binomial coefficients, like .

Polynomials

Division algorithm. Remainder Theorem. Number of roots. Polynomials in . Irreducibles and unique factorization. and Gauss's Lemma. Cyclotomic polynomials.

Orders of elements

Units. The group . Computing orders. Cyclicity of . For which is cyclic?

Quadratic reciprocity

Legendre symbols. Euler's criterion. Gaus's fourth proof of Reciprocity. Jacobi symbols.

Continued fractions

Computing convergents. . Best rational approximations. Pell's equation.

Arithmetic functions

, , , and . Multiplicative functions. Sum of as divides . Möbius Inversion. Convolutions of functions.

Gaussian integers:

Norms. Which rational primes have Gaussian factors? Division algorithm. Unique factorization. Fermat's two squares theorem. Counting residues .

Finite fields

Characteristic. Frobenius map. Counting irreducible polynomials. Uniqueness Theorem for the field of elements.

Resultants

Discriminant of a polynomial and formal derivatives. Resultant of two polynomials and relation with Euclid's algorithm.

Geometry of numbers

Lattice points. Pick's Theorem. Minkowski's Theorem. Geometric interpretation of the Farey sequence and continued fractions. Geometric proofs of the two square and four square theorems.

Quadratic number fields

Which quadratic number rings are Euclidean? For which does have a division algorithm using the norm?