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B018797 - NUMBER THEORY
Main information
Course Content
Suggested readings
Learning Objectives
Prerequisites
Type of Assessment
Course program
Academic Year 2015-16
Coorte 2015 - Second Cycle Degree in MATHEMATICS
Course year
First year -
Belonging Department
Mathematics and Computer Science "Ulisse Dini" (DIMAI)
Course Type
Single education field course
Scientific Area
MAT/02 - ALGEBRA
Credits
9
Teaching Hours
72
Teaching Term
⇒
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Mutuality
Course teached as:
B018797 - TEORIA DEI NUMERI
Second Cycle Degree in MATHEMATICS
Curriculum GENERALE
B018797 - TEORIA DEI NUMERI
Second Cycle Degree in MATHEMATICS
Curriculum GENERALE
Course Content
Number fields. Number rings. Discriminant for a number field. Fractional ideals. Unique factorization for fractional ideals in a number ring. Class group. Minkowski's theorem. Finiteness of the class group. Dirichlet's units theorem. First Case of Fermat's Last Theorem for regular primes. Dirichlet's L-functions. Riemann's zeta function. Dirichlet's theorem. p-adic numbers. Quadratic forms over the p-adics. Integer quadratic forms.
Suggested readings (Search our library's catalogue)
Stewart-Tall: Algebraic number theory. Chapman and Hall Mathematics Series Chapman & Hall, London, 1987.
Marcus: Number Fields Universitext. Springer-Verlag, New York-Heidelberg, 1977
Serre : A course in arithmetic. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973
Marcus: Number Fields Universitext. Springer-Verlag, New York-Heidelberg, 1977
Serre : A course in arithmetic. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973
Learning Objectives
Knowledge acquired:
The aim of the course is to give the fundamental knowledge of algebraic number theory . In a second stage the tools developed are used to prove some important results, e.g. Dirichlet's units theorem, the finiteness of class groups etc.
Competence acquired:
The student who has successfully passed the exam , should have the basic knowledge to understand more advanced topics in number theory.
Skills acquired (at the end of the course):
The student should be able to perform calculations in number fields, manipulating elements as well as ideals.
The aim of the course is to give the fundamental knowledge of algebraic number theory . In a second stage the tools developed are used to prove some important results, e.g. Dirichlet's units theorem, the finiteness of class groups etc.
Competence acquired:
The student who has successfully passed the exam , should have the basic knowledge to understand more advanced topics in number theory.
Skills acquired (at the end of the course):
The student should be able to perform calculations in number fields, manipulating elements as well as ideals.
Prerequisites
Courses required: Algebra I,II
Type of Assessment
Oral exam
Course program
Number fields. Number rings. Discriminant for a number field. Fractional
ideals. Unique factorization for fractional ideals in a number ring. Class
group. Minkowski's theorem. Finiteness of the class group. Dirichlet's
units theorem. First Case of Fermat's Last Theorem for regular primes.
Dirichlet's L-functions. Riemann's zeta function. Dirichlet's theorem. padic
numbers. Quadratic forms over the p-adics. Integer quadratic forms.
ideals. Unique factorization for fractional ideals in a number ring. Class
group. Minkowski's theorem. Finiteness of the class group. Dirichlet's
units theorem. First Case of Fermat's Last Theorem for regular primes.
Dirichlet's L-functions. Riemann's zeta function. Dirichlet's theorem. padic
numbers. Quadratic forms over the p-adics. Integer quadratic forms.