Algebra

Below are the topics for the qualifying exam in Algebra

1. Group Theory
   • Group actions; counting with groups.
   • p-groups and Sylow theorems.
   • Composition series; Jordan-Holder theorem; solvable groups.
   • Automorphisms; semi-direct products.
   • Structure of finitely generated Abelian groups
2. Linear Algebra and Commutative Algebra.
   • Euclidean domain implies PID implies UFD.
   • Gauss Lemma; Eisenstein’s Criterion.
   • Exact sequences; isomorphism theorems for modules.
   • Free modules.
   • Hom and tensor product of vector space, Abelian groups, and modules; Restriction and extension of scalars.
   • Bilinear forms; symmetric and alternating forms; symmetric and exterior algebras.
   • Structure Theorem for finitely generated modules of a PID.
   • Rational canonical form.
   • Jordan canonical form.
   • Chain conditions; Noetherian rings and modules; Hilbert’s Basis Theorem.
   • Prime and maximal ideals.
   • Field of fractions.
   • Localization of rings and modules: exactness of localization; local rings; Nakayama’s Lemma.
   • Integral extensions.
   • Noether’s Normalization Lemma.
   • Integral closure.
   • Nullstellensatz.
   • Closed affine algebraic sets.
3. Field Theory and Galois Theory
   • Algebraic field extensions: finite extensions; degree of extensions; the minimal polynomial; adjoining roots of polynomials; the splitting field; algebraic closure.
   • Separable extensions.
   • Theorem of the primitive element.
   • Galois extensions: Fundamental Theorem of Galois Theory.
   • Finite fields and their Galois groups; Frobenius endomorphism.
   • Cyclotomic polynomial, Cyclotomic fields and their Galois groups.
   • Cyclic extensions.
   • Solvable extensions; Solving polynomial equations in radicals.
   • Transcendence degree.

References

• Dummit and Foote, Abstract Algebra
• Atiyah and MacDonald, Introduction to Commutative Algebra
• Lang, Algebra