Syllabus for Algebra 1 Qualifying Exam

The Algebra 1 exam covers essentially the material taught in MTH G111 (Algebra 1). There will be 10 problems: 5 from Linear Algebra and 5 from Group Theory. Minimum required for passing the exam is 70%.

Linear Algebra

• Vector spaces, bases, matrices, linear maps, kernel, image.

• Scalar products and orthogonality, bilinear maps, dual space, quadratic forms, Sylvester theorem.

• Determinants, inverse of a matrix, rank, subdeterminants.

• Symmetric, Hermitian and unitary operators.

• Eigenvectors, eigenvalues, characteristic polynomial.

• Diagonalization, triangulation, Jordan normal form.

• Tensor products of vector spaces: universality property and construction.

• Exterior and symmetric powers of a vector space, exterior and symmetric multiplication, contractions.

Group Theory

• Groups, subgroups, normal subgroups, cosets, quotient groups, group homomorphisms.

• Isomorphism theorems, actions of groups (G-sets), counting orbits.

• Sylow theory, normal series, solvable groups.

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

References

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

• Serge Lang, Introduction to Linear Algebra, second edition, Springer GTM #211, 1997

• 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)