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Week
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Date
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Topics
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Reading
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Homework
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1
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Tue 15 Jan
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No class!
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Thu 17 Jan
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History of algebraic integers.
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ANT
pp. 5-13
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2
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Tue 22 Jan
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Algebra review: ideal operations, prime and maximal ideals, finitely
generated modules. Integral elements and equivalent conditions.
Integral closure.
|
ANT 1
pp. 14-27
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Thu 24 Jan
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Integrally closed rings. UFD's are integrally closed. Transitivity
of integral closure.
|
ANT 2
pp. 25-30
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3
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Tue 29 Jan
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Free modules. Basis. Norm and trace.
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ANT
pp. 31-34
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Problem Set #1
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Thu 31 Jan
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More norm and trace. Discriminant.
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ANT
pp. 31-34
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4
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Tue 05 Feb
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More discriminant. Discriminant and bases. Discriminant and separability.
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ANT
pp. 33-34
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Thu 07 Feb
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Finite generation of the integral closure. Integral basis.
Computing the ring of integers.
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ANT
pp. 35-36
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5
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Tue 12 Feb
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More on computing the ring of integers. Stickelberger's theorem. Taussky-Todd's theorem.
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ANT
pp. 37-43
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Problem Set #2
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Thu 14 Feb
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Dedekind domains. Integral closures of Dedekind domains. Unique factorization into prime ideals.
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ANT
pp. 45-51
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6
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Tue 19 Feb
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Unique factorization into prime ideals. Invertible ideals.
Fractional ideals. Class group.
|
ANT
pp. 45-51
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Thu 21 Feb
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Generation by two elements. Factorization of primes.
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ANT
pp. 51-58
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7
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Tue 26 Feb
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Ramification index and inertial degree.
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ANT
pp. 57-59
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Thu 28 Feb
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Ramified primes and the discriminant.
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ANT
pp. 59-61
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8
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Tue 05 Mar
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Explicit methods for factorization of primes.
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ANT
pp. 61-64
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Thu 07 Mar
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Geometry of numbers. Minkowski bound. Computing class groups.
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ANT
pp. 69-80
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9
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Tue 12 Mar
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Spring Break!
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Thu 14 Mar
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Spring Break!
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10
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Tue 19 Mar
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Spring Break!
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Thu 21 Mar
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Spring Break!
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11
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Tue 26 Mar
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Finiteness of class group.
|
ANT
pp. 79-80
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Midterm
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Thu 28 Mar
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More computations of class groups.
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ANT
pp.
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12
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Tue 02 Apr
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Dirichlet Unit Theorem. Fundamental units. Beginnings of proof.
Logarithmic embedding. Roots of unity. Finite generation of unit group.
|
ANT
pp. 84-86
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Thu 04 Apr
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Bounds on fundamental unit in real quadratic fields. Verifying a
fundamental unit.
|
ANT
pp.
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13
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Tue 09 Apr
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Artin's bound on fundamental unit in real cubic fields.
|
ANT
pp. 91-92
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Thu 11 Apr
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Eisenstein criterion for computing the ring of integers.
Midterm exam!
|
ANT
pp.
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14
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Tue 16 Apr
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Using fundamental unit in real cubic field to prove nontriviality of
class group elements.
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Thu 18 Apr
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Topological proof of unit theorem.
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15
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Tue 23 Apr
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Topological proof of unit theorem, continued.
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Conrad's Notes
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Thu 25 Apr
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Topological proof of unit theorem, end.
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16
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Mon 29 Apr
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Reading week: Diophantine problems.
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Wed 01 May
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Pythagorean triples. Fermat's Last Theorem for regular primes. Farewell!
|
ANT
pp. 100-103
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Wed 08 May
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Final exam!
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Final exam
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