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Week
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Date
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Topics
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Resources
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1
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Wed 29 Aug
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Organizational meeting.
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Fri 31 Aug
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Introduction. Reminders on fields and Galois theory. Profinite groups.
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Gille-Szamuely Appendix A.1
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2
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Mon 03 Sep
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Labor Day!
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Wed 05 Sep
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Infinite Galois theory.
Projective
space and algebraic varieties. Functor of points.
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Gille-Szamuely Appendix A.1
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3
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Mon 10 Sep
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Quaternion algebras. Norm form.
Wedderburn's theorem characterizing quaternion algebras.
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Lam III.1,
Gille-Szamuely 1.1-1.2
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Wed 12 Sep
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Conics. Quadratic subfields.
Hilbert symbol.
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Gille-Szamuely 1.1-1.3, John Tate's Symbols
in Arithmetic
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4
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Mon 17 Sep
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Class cancelled.
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Wed 19 Sep
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Class cancelled.
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5
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Mon 24 Sep
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Central simple algebras. Artin-Wedderburn theory.
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Gille-Szamuely 2.1, 2.4
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Wed 26 Sep
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Brauer group. Period/index/degree.
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Gille-Szamuely 2.1, 2.4
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6
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Mon 01 Oct
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Tensor products of algebras. Biquaternion algebras. Albert form.
Examples of Amitsur-Rowen-Tignol. Merkurjev's theorem. Milnor K-theory
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Gille-Szamuely 1.5
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Wed 03 Oct
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Milnor K-theory. Graded commutativity. Importance of Steinberg relation.
Reinterpretation of Merkurjev's theorem.
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Gille-Szamuely 7.1
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7
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Mon 08 Oct
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Computations of some Milnor K-theory groups. Cyclic algebras.
Merkurjev-Suslin. The category of discrete modules over a group.
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Gille-Szamuely 7.1, 2.5, 3.1
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Wed 10 Oct
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Classical group cohomology. Review of derived functors and Ext.
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Gille-Szamuely 3.1-3.2, Serre I.1.1
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8
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Mon 15 Oct
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Standard resolution. Inhomogeneous cochains. Group extensions.
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Gille-Szamuely 3.2, Shatz I.IV.1,
Serre I.2.1, II.2.2, Shatz II.1
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Wed 17 Oct
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Fall Break!
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9
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Mon 22 Oct
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Finite additive, multiplicative, and classical Hilbert's Theorem 90.
Discrete modules.
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Gille-Szamuely 2.3.4,
Serre I.2.1, II.2.2, Shatz II.1
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Wed 24 Oct
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Profinite group cohomology via derived functors, inhomogeneous
cocycles, and limits.
Functoriality. Restriction. Inflation.
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Gille-Szamuely 3.3, 4.1-2,
Serre I.2.1, II.2.2, Shatz II.1
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10
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Mon 29 Oct
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Profinite Hilbert's Theorem 90.
Roots of unity. Cohomological and classical Kummer theory. Kummer
sequence. Coboundary maps in the long exact sequence.
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Gille-Szamuely 4.3
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Wed 31 Oct
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Kummer theory examples. Classification of cyclic extensions.
Cohomological and classical Artin-Schreier theory. Artin-Schreier sequence.
Nonabelian H1. Crossed homomorphisms. Pointed sets.
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Gille-Szamuely 2.3, 4.3.
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11
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Mon 05 Nov
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Etale algebras. Galois cohomology of the symmetric group.
Grothendieck's Galois theory
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Wed 07 Nov
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Galois descent. Forms of tensors. Classification via nonabelian H1
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Gille-Szamuely 2.3, Serre III.1.1, I.5.2
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12
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Mon 12 Nov
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Generalized Hilbert's Theorem 90. Proof of Galois descent. Twisting.
Long(ish) exact sequence in nonabelian cohomology.
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Gille-Szamuely 2.3, 2.7, Serre III.1.1, I.5.2, I.5.4
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Wed 14 Nov
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Proof of the long(ish) exact sequence.
Cohomological Brauer group.
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Gille-Szamuely 4.4-4.5
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13
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Mon 19 Nov
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Thanksgiving Break.
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Wed 21 Nov
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Thanksgiving Break.
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14
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Mon 26 Nov
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More cohomological Brauer group. Finiteness of period. Period-index problem.
Reminders about p-adic numbers.
Extensions of discrete valuations.
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Gille-Szamuely 4.4-4.5, A.6
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Wed 28 Nov
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Ramification in extensions of complete discretely valued fields.
Unramified extensions and extensions of the residue field. Maximal
unramified extension.
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Serre LF II.3, III.5
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15
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Mon 03 Dec
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Existence of unramified splitting fields of central simple algebras
over complete discretely valued fields. Brauer group of complete
discretely valued fields. Hasse invariant of local field. Statement of
Hasse-Brauer-Noether exact sequence for global field.
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Serre LF XII.1-3, Gille-Szamuely 6.5
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Wed 5 Dec
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Multivariable Hensel's lemma.
Hasse-Brauer-Noether exact sequence, Hilbert reciprocity, and
quadratic reciprocity. Hasse Principle for varieties over number
fields: conics, zero-dimensional varieties, statement of
Hasse-Minkowski, example of Selmer.
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Serre LF XII.1-3, Gille-Szamuely 6.5,
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16
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Mon 10 Dec
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Reading Period Bonus Class. Proof of the
Hasse-Minkowski theorem. Some quadratic forms theory. Global
squares theorem and Chebotarev density. Weak approximation and norms.
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Scharlau 6.6
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Wed 12 Dec
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Reading Period.
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