Participants / Math at Ross

# 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?