Entry requirements

Algebra II

Learning outcomes

In order to pass the course (grade 3) the student should be able to

• compute the greatest common divisor by help of Euclid's algorithm;
• give an account of the proof of the fundamental theorem of arithmetic;
• solve linear Diophantine equations;
• solve linear congruences;
• solve polynomial congruences by help of the Chinese remainder theorem and Hensel's lemma;
• be familiar with the Möbius inversion formula;
• be familiar with the parametrisation of primitive Pythagorean triples;
• decide if a given number is the sum of two or three squares;
• decide whether a given number is a quadratic residue modulo p;
• develop a quadratic irrational number into continued fractions;
• decide the value of a periodical continued fraction;
• solve Pell's equation.

Content

Divisibility; ideals in the ring of integers, Euclid's algorithm, the fundamental theorem of arithmetic. Linear Diophantine equations, groups of units in quotients of the ring of integers, the Chinese remainder theorem, Hensel's lemma. Cyclic groups of units and primitive roots, order. Quadratic residues and quadratic reciprocity. Arithmetic functions and the Möbius inversion formula. Sums of squares, Pythagorean triples. Continued fractions, rational approximation. Pell's equation.

Instruction

Lectures and problem solving sessions.

Assessment

Written and possibly oral examination at the end of the course combined, if required, with written assignments during the course according to instructions provided at course start.