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Quick Facts
| Medium Of Instructions | Mode Of Learning | Mode Of Delivery |
|---|---|---|
| English | Self Study | Video and Text Based |
Important dates
Course Commencement Date
Start Date : 20 Jan, 2025
End Date : 11 Apr, 2025
Enrollment Date
End Date : 27 Jan, 2025
Certificate Exam Date
Start Date : 26 Apr, 2025
Other
End Date : 14 Feb, 2025
Courses and Certificate Fees
| Fees Informations | Certificate Availability | Certificate Providing Authority |
|---|---|---|
| INR 1000 | yes | IISc Bangalore |
The Syllabus
- Study of number fields, definition of the ring of integers. Definition of norm and trace.
- Definition of absolute and relative discriminant. Computation of discriminant. Computation of the ring of integers.
- Definition and properties of Dedekind domains. Proof that the ring of integers is a Dedekind domain. Factorisation of extension of prime ideals in a finite extension of number fields.
- Embeddings of a number field in complex numbers. A result from geometry of numbers. Finiteness of class groups.
- Computation of class groups, including several examples. Applications to Diophantine equations of computations of class groups.
- Dirichlet’s unit theorem.
- Extension and norm of ideals in field extensions. Maps between class groups of extensions. Decomposition subgroups, inertia subgroups, Frobenius elements etc. Localisation, residue field.
- Valuations in a number fields. Local fields. Hensel’s lemma and applications.
- Field extensions of local fields, ramification, different, inertia subgroups etc.
- Study of special number fields. Imaginary quadratic fields, real quadratic fields, cubic fields, cyclotomic fields.
- Definition of ray class field as a generalisation of ideal class group. Some statements from class field theory without proofs.
- Definition of zeta functions and L-functions. Statements of their analytic properties without proofs. Dirichlet Class number formula.
Instructors
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