Algebra Qualifying Examination Syllabus

Linear Algebra: Bases, dimension, dual spaces. Matrices and linear transformations. Minimal and characteristic polynomials. The theory of a single linear transformation including both the rational and Jordan cannonical forms.Group Theory: Groups acting on a set. Normal subgroups and the homomorphism theorems. Lagrange's Theorem: Sylow theory. Structure theory for finitely generated Abelian groups.Rings and Fields: Unique factorization domains and principal ideal domains. Ideals, homomorphisms. Finite fields, algebraic number fields, subfields of the complex numbers. Normal and separable extension field and Galois theory.The reference below each contains the material of the syllabus. The chapters indicating where the material can be found.

  1. B. L. van der Waerden. Modern Algebra (in two volumes). Revised English Edition. Chapters I-VII, XV. This is a classic text written in 1930. Although the notation is not in common use today the material is treated in a way that still could be called modern.
  2. MacLane and Birkhoff, Algebra (second edition). Chapters I-III, V-VII, XI-XIII. This is a thorough revised edition of the classic "A survey of modern algebra" by Birkhoff and MacLane.
  3. Nathan Jacobsen, Basic Algebra I. Chapters 0-4. This is a two volume revision of the authors three volume classic. Although the first volume contains the material of the syllabus the student may want to dip into the second volume also.
  4. Serge Lang, Algebra Chapters I-III, V, VII, VIII, XV
  5. Hungerford, Algebra Chapters I-VII