Syllabus for Algebra 2 Qualifying Exam

The Algebra 2 exam covers essentially the material taught in MTH G112 (Algebra 2). There will be 10 problems. Minimum required for passing the exam is 70%.

Groups

• Groups, subgroups, normal subgroups, cosets of a subgroup, quotient groups, exact sequences of groups.

• Sylow theorems and solvable groups.

• Examples: cyclic groups, symmetric groups, direct and semidirect products.

• Abelian groups and their homomorphisms, exact sequences of abelian groups.

• Classification of finite abelian groups, infinite abelian groups, free abelian groups, torsion and torsion-free abelian groups, infinite products and sums.

Rings

• Rings, ideals, commutative rings, subrings and quotient rings.

• Units and factorization. polynomial rings.

• Fields and finite fields.

• Domains, Euclidean domains, principal ideal domains.

• Unique factorization domains and the Gauss lemma, Unique factorization in polynomial rings.

Modules

• The category of modules over a ring, submodules and quotient modules, exact sequences and relations with Hom.

• Cyclic modules, free modules and projective modules.

• Linear algebra of Euclidean rings and principal ideal domains.

• Abelian groups as modules, structure of finitely generated abelian groups and of finitely generated modules over a principal ideal domain.

• Relations of modules to problems of factorization.

References

• Serge Lang, Algebra, revised third edition, Springer GTM #211, 2002 (Chapter 1)

• Thomas W. Hungerford, Algebra, Springer GTM #73 (Chapters I-IV)

• George D. Mostow, Joseph H. Sampson, Jean-Pierre Meyer, Fundamental Structures of Algebra, McGraw-Hill, New York 1963.

• Joseph J. Rotman, An Introduction to the Theory of Groups, Springer GTM #148 (Chapters 1-6 and 10)