URL study guide
https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=2MBD20&collegejaar=2025&taal=enOmschrijving
Algebra aims at understanding computational structures from an abstract unified point of view, so as to apply the results in a variety of settings, ranging from theoretical settings (extending our knowledge of mathematics) to practical settings (like coding theory and cryptography, algebraic vision, or games like Rubik's cube). This leads to the course’s focus on algebraic concepts like groups and rings, where the main ingredients are a set and one or more operations obeying certain rules. For instance, the ordinary integers together with the operation `plus' (obeying certain properties, like commutativity) is an example of a group; the same ordinary integers with the two operations `plus' and `times' (again, with certain properties) is an example of a ring.We will focus on (concepts and results):
• polynomials (univariate, division with remainder, gcd, multivariate, (leading) monomials)
• permutations and permutation arithmetic (cycles, conjugation, cycle structure, sign of a permutation)
• algebraic structures: emphasis will be on (axioms for) groups and rings and their substructures;
• ideals: principal, prime and maximal ideals; generators for ideals;
• special structures like cyclic groups, (normal) subgroups, subrings, abelian groups, integral domains;
• constructions with the structures, like product groups, quotient groups and quotient rings;
• group and ring homomorphisms, i.e., special maps between groups, rings, etc., and results on homomorphisms (like isomorphism, injective, surjective homomorphism, isomorphism theorems) in the setting of these structures; theorems, etc., on groups and rings, such as Lagrange’s theorem, classification of cyclic groups, the relation between prime (resp. maximal) ideals and integral domains (resp. fields).
• computations within the various structures (such as Gröbner basis computations);
• actions of groups on (structured) sets
Doelstellingen
The student knows the various concepts, techniques and basic results concerning groups and rings, their relevance for mathematics, and is able to• carry out meaningful computations in and with such structures, including
some algorithmic aspects;
• use and prove results (theorems, computational techniques) on such structures.