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B005501 - ALGEBRA III
Main information
Course Content
Suggested readings
Further information
Type of Assessment
Course program
Academic Year 2015-16
Coorte 2013 - 3-years First Cycle Degree (DM 270/04) in MATHEMATICS
Course year
Third year - Second Semester
Belonging Department
Mathematics and Computer Science "Ulisse Dini" (DIMAI)
Course Type
Single education field course
Scientific Area
MAT/02 - ALGEBRA
Credits
6
Teaching Hours
48
Teaching Term
01/03/2016 ⇒ 17/06/2016
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Course Content
GALOIS THEORY
COMMUTATIVE ALGEBRA
COMMUTATIVE ALGEBRA
Suggested readings (Search our library's catalogue)
I. M. Isaacs, Algebra: a graduate course, Brooks/Cole Publishing Company, 1994.
M. Atiyah, I. Macdonald, Introduction to Commutative Algebra; Addison-Wesley Publishing Company, 1969.
M. Atiyah, I. Macdonald, Introduction to Commutative Algebra; Addison-Wesley Publishing Company, 1969.
Further information
Attending lectures: recommended.
Teaching Tools UniFi e-Learning: http://e-l.unifi.it
Office hours:
By appointment.
Department of Mathematics "Ulisse Dini"
Viale Morgagni, 67/a
50134 - Firenze (FI)
Tel: 055 2751443
email: dolfi@math.unifi.it
Teaching Tools UniFi e-Learning: http://e-l.unifi.it
Office hours:
By appointment.
Department of Mathematics "Ulisse Dini"
Viale Morgagni, 67/a
50134 - Firenze (FI)
Tel: 055 2751443
email: dolfi@math.unifi.it
Type of Assessment
Oral
Course program
(I) Galois Theory
Survey of basic Galois Theory: algebraic field extensions, normal extensions, separable extensions.
Splitting fileds. Galois correspondence.
Artin's theorem. Primitive element theorem. Dedekind's theorem (linear independence of field automorphsims). Normal basis theorem. Natural irrationalities theorem. Algebraic closures.
Cyclotomic fields. Gauss' theorem.
Purely inseparable field extensions.
Automorphism groups of cyclic groups. Structure of finitely generated abelian groups.
Dirichlet's theorem.
Series in groups. Solvable groups. Simplicity of alternating groups A_n, n >4.
Kummer's theory.
Solvability of equations by radicals.
(II) Commutative Algebra
Rings; Modules.
Exactness. Tensor products of modules and algebras.
Restrictions and extensions of ideals. Torsion free modules; projective modules. Modules over PID.
Rings and modules of fractions.
Primary decomposition of ideals.
Integral dependence. Going-up and going-down theorems. Integral basis theorem.
Chain conditions. Noetherian rings. Hilbert basis theorem.
Artinian rings.
Discrete valuation rings. Dedekind rings. Ideal decomposition in Dedekind rings.
Survey of basic Galois Theory: algebraic field extensions, normal extensions, separable extensions.
Splitting fileds. Galois correspondence.
Artin's theorem. Primitive element theorem. Dedekind's theorem (linear independence of field automorphsims). Normal basis theorem. Natural irrationalities theorem. Algebraic closures.
Cyclotomic fields. Gauss' theorem.
Purely inseparable field extensions.
Automorphism groups of cyclic groups. Structure of finitely generated abelian groups.
Dirichlet's theorem.
Series in groups. Solvable groups. Simplicity of alternating groups A_n, n >4.
Kummer's theory.
Solvability of equations by radicals.
(II) Commutative Algebra
Rings; Modules.
Exactness. Tensor products of modules and algebras.
Restrictions and extensions of ideals. Torsion free modules; projective modules. Modules over PID.
Rings and modules of fractions.
Primary decomposition of ideals.
Integral dependence. Going-up and going-down theorems. Integral basis theorem.
Chain conditions. Noetherian rings. Hilbert basis theorem.
Artinian rings.
Discrete valuation rings. Dedekind rings. Ideal decomposition in Dedekind rings.