{"id":5711424,"date":"2021-11-01T20:47:11","date_gmt":"2021-11-02T06:47:11","guid":{"rendered":"https:\/\/math.hawaii.edu\/wordpress\/?p=5711424"},"modified":"2024-06-19T17:27:43","modified_gmt":"2024-06-20T03:27:43","slug":"qualifying-exams-in-algebra","status":"publish","type":"post","link":"https:\/\/math.hawaii.edu\/wordpress\/qualifying-exams-in-algebra\/","title":{"rendered":"Qualifying exams in algebra"},"content":{"rendered":"
Algebra exam<\/h5>\n

The algebra qualifying exam covers several standard topics in abstract algebra.<\/p>\n

Material<\/h6>\n
    \n
  1. Group theory:<\/strong> basics of group actions, semidirect products, class equation, Sylow theorems, applications, solvable groups, Jordan\u2013H\u00f6lder theorem<\/li>\n
  2. Field and Galois theory:<\/strong> finite fields, separable and normal extensions, Fundamental theorem of Galois theory, applications (e.g. solvability by radicals, constructions by straightedge and compass, …), determining Galois groups<\/li>\n
  3. Ring theory:<\/strong> factorization in domains, simplicity of matrix algebras<\/li>\n
  4. Module theory:<\/strong> basics, projectivity, injectivity, tensor products, flatness, Noetherian property, exact sequences, commutative diagrams, structure theory of modules over a PID, consequences for canonical forms of matrices and other linear algebra<\/li>\n
  5. Language of category theory:<\/strong> objects, arrows, Hom, functors, natural transformations, universal objects, products, coproducts, Yoneda lemma<\/li>\n
  6. Multilinear algebra:<\/strong> pairings, wedge products, symmetric products, multilinear forms over rings<\/li>\n
  7. Basic commutative algebra:<\/strong> local rings and localization, integral extensions, Hilbert Basis Theorem, Noether Normalization, Hilbert\u2019s Nullstellensatz<\/li>\n<\/ol>\n
    Textbooks<\/h6>\n

    The following textbooks are recommended:<\/p>\n