Algebra 1 (Fall 2025)

Syllabus

• Number of contact hours: lecture: 3, problem session: 2
• Number of credits: 7
• Course schedule:
  Lecture: Thursday 12.15–15.00, R507
  Problem session: Monday 14.15–16.00, H601
• Instructors:
  Erzsébet Lukács, lukacs@math.bme.hu
  Balázs Mezei, balazs.f.mezei@gmail.com
• Webpage: www.math.bme.hu/~lukacs/bboard/alg1/2025/
• Prerequisites: Vector and Matrix algebra and Introduction to Algebra
• Textbook: none
• Recommended reading
  Isaacs: Algebra. A Graduate Course
  Herstein: Abstract algebra
• Grading: Homework + two midterm tests + written exam, maximum 200 points:
  Homework (30 points)
  Tests (35 points each)
  Exam (100 points)
  For a passing grade you need at least 40% of the 100 points for term work (and in it at least 40%, that is, 14 points from one of the tests), and at least 40% at the exam. There will be one make-up test (also for improving the term grade) in the last week of the term, which can overwrite one of the test results. The lower limits for the grades are then 40%, 53%, 66%, 79% for 2,3,4 and 5.
• 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: orbit-stabilizer theorem, counting arguments, simplicity of alternating groups. p-groups, Sylow theorems, applications of the Sylow theorems.
  Rings and fields. Examples, subrings, ideals (left, right, twosided), quotient rings, principal ideals, PID's, euclidean domains, the polynomial ring K[x]. Field extensions, the structure of simple (algebraic, transcendental) extensions. Finite fields.