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Algebraic Curves Database

PhD qualifying exams

There are three qualifying exams for students working in algebra.  
  • Groups and Rings:    PhD qualifying exam in algebra (based on MTH 571)
  • Field and Galois Theory:  PhD qualifying exam in algebra (based on MTH 572)
  • One of the following:   
    • Computational Algebra     (MTH 577)
    • Commutative Algebra       (MTH  671-672)
    • Coding Theory and Cryptography   (MTH 673) 
The exam is graded Pass/Fail and is administered by the Algebra Committee which contains three faculty members of the Department from the area of research in Algebra/Algebraic geometry.  The student can take each exam only twice. 

The PhD exams in algebra are offered every year, during the second half of the months of May and August.  Students who intend to take the exam should register with the graduate committee. 

  • June 2: MTH 571
  • June 9: MTH 572

  • August 18: MTH 571
  • August 25: MTH 572 

Groups and Rings. 

The exam is based on the contents of the sequence MTH 571. However, topics that are not covered in the course can also be on the exam.  As textbooks recommended for the study among others are
  • An introduction to the theory of groups, J. Rotman
  • Finite Group Theory, I. Martin Isaacs
  • Algebra, Lang
  • Algebra, Hungerford

Topics

This examination will cover topics listed as below

Group Theory

    • Sylow theorems
    • Solvable and nilpotent groups
    • Abelian groups
    • p-groups
    • nilpotent groups
    • Linear Groups

Rings

  • Basics on rings
  • Polynomial Rings
  • Localization
  • Notherian rings; the Hilbert Basis Theorem

Commutative Algebra

The exam is based on the contents of the sequence MTH 671. However, topics that are not covered in the course can also be on the exam.  As textbooks recommended for the study are

Books

  • Introduction to Commutative Algebra, Atyah & MacDonald. 
  • Algebra, Lang
  • Algebra, Hungerford

Topics 

Homological Algebra

  • Categories and functors
  • Products and coproducts
  • Equilazers, pushbacks, pushouts
  • Limits and colimits
  • Inverse limits

Module theory

    • Modules
    • Modules over PID's
    • Direct sums and free modules
    • Exact sequences
    • Free modules; Projective and Injective modules
    • Tensor Products
    • Linear algebra
    • Canonical forms
    • Artinian Rings; the Wedderburn-Artin Theorem


Fields and Galois Theory. 

The exam is based on the contents of the sequence MTH 572. However, topics that are not covered in the course can also be on the exam.  As textbooks recommended for the study are

Books

  • Algebra, S. Lang
  • Field Theory, Roman
  • Field and Galois Theory, Morandi

Topics

  • Field Extensions
  • Algebraic Extensions
  • Separable Extensions
  • Norms and Traces
  • Galois Theory
  • Galois grous of polynomials
  • Abelian Extensions
  • Finite Fields
  • Transcendental Extensions



Computational Algebra

This Exam will be first offered in the Summer 2013.  The list of topics is subject to change

Topics


  • Computational Group Theory
  • Algebraic Equations
  • Invariant Theory
  • Commutative algebra and Algebraic Geometry

A more detailed list will be available soon.



Coding Theory

The coding theory Exam will first be offered in the Summer 2013. The exam is based on the following book

Books

  • Fundamental of Error Correcting Codes, C. Huffman and V. Pless

Topics

All basic topics of the book might be covered in the exam.  
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