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Week
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Date
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Topics
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Resources
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2
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Thu 28 Jan
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Introduction. Quaternion algebras.
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Lam III.1,
Gille-Szamuely 1.1-1.2
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3
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Tue 02 Feb
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Quadratic subfields. Conics.
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Gille-Szamuely 1.2-1.3
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Thu 04 Feb
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Hilbert symbol. Specialization techniques from k(t).
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Gille-Szamuely 1.3
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4
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Tue 09 Feb
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Absolute values and valuations. Valuation rings. Discrete valuations.
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Gille-Szamuely A.6
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Thu 11 Feb
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Ostrowski's Theorem for Q and
k(t).
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K. Conrad's Ostrowski's Theorem over Q
and k(t)
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5
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Tue 16 Feb
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Tame symbol. Central simple algebras. Artin-Wedderburn. Brauer
group. Period/index/degree.
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John Tate's Symbols
in Arithmetic, Gille-Szamuely 2.1, 2.4
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Thu 18 Feb
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Biquaternion algebras. Albert form. Examples of Amitsur-Rowen-Tignol. Milnor K-thoory.
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Gille-Szamuely 1.5, 7.1
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6
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Tue 23 Feb
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More Milnor K-theory. Examples: finite fields, R. Cyclic
algebras and symbols. Theorems of
Merkurjev and Merkurjev-Suslin.
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Gille-Szamuely 7.1
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Thu 25 Feb
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Galois descent. Nonabelian cohomology. Twisted forms. Hilbert 90.
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Gille-Szamuely 2.3, Serre III.1.1, I.5.2
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7
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Tue 01 Mar
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Proof of Galois descent. Twisting.
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Gille-Szamuely 2.3, Serre III.1.1, I.5.2
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Thu 03 Mar
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Reduced norm and trace of central simple algebra.
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Gille-Szamuely 2.6
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8
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Tue 08 Mar
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Exact sequence in nonabelian cohomology.
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Gille-Szamuely 2.7, Serre I.5.4
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Thu 10 Mar
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Applications: Skolem-Noether, cohomology of SL1(A).
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Gille-Szamuely 2.7, Serre I.5.4
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9
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Tue 15 Mar
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Spring Break!
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Thu 17 Mar
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Spring Break!
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10
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Tue 22 Mar
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Spring Break!
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Thu 24 Mar
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Spring Break!
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11
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Tue 29 Mar
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Profinite groups. Open and closed subgroups. Infinite Galois theory.
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Gille-Szamuely 4.1, Serre I.1.1. Shatz I, IV.1.
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Thu 31 Mar
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Discrete modules. Profinite group cohomology via derived functors.
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Serre I.2.1, Shatz II.1,
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12
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Tue 05 Apr
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Inhomogeneous cochains. Homogeneous cochains.
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Gille-Szamuely 4.2, Serre II.2.2, Shatz II.1
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Thu 07 Apr
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Finite group cohomology. Standard resolution. Homotopy retractions.
Explanation of inhomogneous cochains.
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Gille-Szamuely 3.1-3.2, 4.2
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13
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Tue 12 Apr
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Restriction and induction as adjoint functors. Shapiro's Lemma.
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Gille-Szamuely 3.3,4.1, Serre I.2.4-2.5, Shatz II.2-3
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Thu 14 Apr
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Restriction-corestriction sequence. Kummer theory.
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Gille-Szamuely 4.3, Serre I.2.4-2.5, Shatz II.2-3
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14
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Tue 19 Apr
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Class canceled.
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Thu 21 Apr
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Class canceled.
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15
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Tue 26 Apr
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Cohomological Brauer group.
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Gille-Szamuely 4.4-4.5
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Thu 28 Apr
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BATMOBYLE
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14
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Tue 3 May
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Reading period.
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Thu 5 May
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Algebraic dimension of fields. Ci fields.
Cohomological dimension. Chevalley's theorem. Tsen's theorem and results of Lang.
Questions of Artin. Results of Ax. Interpretation in terms of
algebraic geometry and rational connectivity.
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Gille-Szamuely 4.3, Serre I.2.4-2.5, Shatz II.2-3
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