Algebra 1 (Fall 2026)

Syllabus

• Number of contact hours: lecture: 3, problem session: 2
• Number of credits: 7
• Course schedule:
  Lecture: Thursday 10.15–11.45 in H601, continuing at 12.10–12.55 in H401
  Problem session: Wednesday 14.15–15.45, E502
• Instructor: Erzsébet Lukács, lukacs@math.bme.hu
• Webpage: www.math.bme.hu/~lukacs/bboard/alg1/2026/
• Prerequisites: Vector and Matrix algebra and Introduction to Algebra
• Textbook: none
• Recommended reading
  Herstein: Abstract algebra
  Isaacs: Algebra. A Graduate Course
• Grading: There will be two written midterm tests consisting of problem solving, and an oral exam in the exam period (the latter including definitions, theorems, examples, problems from the topics of the last two problem sheets not covered in the midterms, and proofs of theorems). You need to pass both midterms to be eligible for taking the final exam (there will be a make-up test in the end of the term where you may improve the marks for one or both tests), and then get a passing mark at the exam. The term work and the exam counts equally in the final grade.
• Topics: Group theory. Examples: groups of symmetries, permutations, matrix groups, abstract definition. Subgroups: generated subgroups, cyclic groups, order of elements, Lagrange's theorem. Homomorphisms: normal subgroups, characterizations, conjugation, quotient groups, direct product, fundamental theorem of finite abelian groups. Permutation groups and group actions: orbit-stabilizer theorem, orbit-counting, calculations in S_n, simplicity of alternating groups. p-groups and p-subgroups, Sylow's theorems, applications of Sylow's theorems.
  Rings and fields. Examples, subrings, ideals (left, right, twosided), quotient rings, principal ideals, PID's, euclidean domains, UFD-s, the polynomial ring K[x]. Field extensions, the structure of simple (algebraic, transcendental) extensions. Finite fields.