Qualifying Exam Syllabus

William Stein

Committee: R. Hartshorne (Chair), H.W. Lenstra, V. Serganova, Bin Yu (Statistics)

1 Algebraic Number Theory

References: Cassels-Frohlich Algebraic Number Theory, Ch I-II; Lang Algebraic Number Theory, Ch VIII.

• Algebraic Number Fields.

– Objects attached to a number field K: discriminant disc(K), different, ring of integers, units, real and complex places, ideal class group, adeles A_{K}, ideles J_{K}, archimedian and nonarchimedian valuations, Riemann’s zeta function.

– Standard facts: the ring of integers is a Dedekind domain, Dirichlet’s unit theorem, the class group is finite.

– Applications of standard tools (not proofs): Analytic class number formula, Chebetarov’s Density Theorem, Kronecker-Weber theorem, Class Field Theory.

• Local Fields.

– Structure: unramified, tamely ramified, and totally ramified extensions, Frobenius automorphisms.

– Standard facts: ef = n, Hensel’s lemma, finite unramified extensions of Q_{p} correspond bijectively to finite extensions of F_{p}, the Brauer group of a local field.

• Examples. Q, Q(\sqrt{D}), Q(ζ_{p}), cubic fields, biquadratic fields. Explicit computation of discriminant, different, class group, units, integers, and galois group of a number field of low degree.

• Applications. Basic coding theory and cryptography.

2 Algebraic Geometry

Reference: Hartshorne, Algebraic Geometry, Ch I-II.

• Varieties over an algebraically closed field. affine and projective varieties, coordinate rings, function fields, rational maps, nonsingular curves, Nullstellensatz.

• Schemes.

– Basics: sheaves, spectrum of a ring, affine schemes, schemes, morphisms between schemes.

– Properties of Schemes: regular, normal, integral, reduced, irreducible, noetherian.

– Properties of Morphisms: separated, finite type, finite, affine, proper, flat.

– Sheaves: coherent and quasi-coherent sheaves, invertible sheaves, the functors f^{∗} (pullback), f_{∗} (pushforward).

– Divisors: Picard group, Cartier divisors, Weil divisors.

3 Lie Algebras

References: Humphreys, Lie Algebras and Representation Theory, ch I-III

– Basic definitions: Lie algebra, the classical Lie algebras, derived series, lower central series, radical.

– Properties of Lie algebras: solvable, nilpotent, semisimple, simple.

– Theorems: Lie’s theorem, Engel’s theorem, Cartan’s criterion.

– Correspondence: the correspondence between simple Lie algebras and irreducible root systems, Killing form, Cartan decomposition.

– Classification of simple root systems: A_{n}, B_{n}, C_{n}, etc.