Post by camilo on Nov 11, 2011 18:13:35 GMT -5
Professor Yotov sent me the email about his proposal for the graduate number theory class and Im so excited! This is as close to dreams coming true I've ever had, ommaggaa!!
Course Proposal: MAS 53XX Algebraic Number Theory
Prerequisites: MAS 4213 Number Theory, MAS 4301 Algebraic Structures
Textbooks: Borevich Z. and I. Shafarevich “Number Theory”
Neukirch J. “Algebraic Number Theory”
Janusz G. “Algebraic Number Fields”
Synopsis:
The course introduces methods in Number Theory developed to study classical problems the area such as representations of rational numbers by (homogeneous) forms and solving Diophantine equations. The observation that a necessary condition for solvability of these problems is their solvability modulo all exponents of any prime number p, leads to introducing the p-adic numbers. The relationship between the field of p-adic numbers, the real numbers and the rational numbers is established by Ostrowski’s theorem: the former fields are all possible completions of the latter one. Arithmetic applications of the p-adic numbers include Hasse-Minkowski’s theorem and the classification of non-singular quadratic forms with rational coefficients. The course next represents the theory of forms with integer coefficients. This leads to introducing decomposable forms as well as modules, orders and maximal orders of a number field. With the use of Minkowski’s geometric methods and his Lemma on Convex Bodies, the Dirichlet’s theorem on the group of units of an order of a number field is proved, and then used to classify, up to similarity, all full modules of that field. The developed general theory is thoroughly discussed in the case of quadratic forms and the related quadratic number fields.
The course next covers in significant detail the topic which led to the development of modern Algebra: the divisibility and unique factorization in number fields. Starting with Euclidean rings, which have unique factorization, the course goes on developing the theory of divisors and valuations, degree of inertia and ramifications of prime divisors. The important Dedekind rings are introduced and studied. This includes divisors and ideals, fractional divisors and the divisor class group of such rings. The results are applied to some special cases of the Fermat’s Last Theorem. Once more the general theory is discussed in detail for quadratic fields.
Course Proposal: MAS 53XX Algebraic Number Theory
Prerequisites: MAS 4213 Number Theory, MAS 4301 Algebraic Structures
Textbooks: Borevich Z. and I. Shafarevich “Number Theory”
Neukirch J. “Algebraic Number Theory”
Janusz G. “Algebraic Number Fields”
Synopsis:
The course introduces methods in Number Theory developed to study classical problems the area such as representations of rational numbers by (homogeneous) forms and solving Diophantine equations. The observation that a necessary condition for solvability of these problems is their solvability modulo all exponents of any prime number p, leads to introducing the p-adic numbers. The relationship between the field of p-adic numbers, the real numbers and the rational numbers is established by Ostrowski’s theorem: the former fields are all possible completions of the latter one. Arithmetic applications of the p-adic numbers include Hasse-Minkowski’s theorem and the classification of non-singular quadratic forms with rational coefficients. The course next represents the theory of forms with integer coefficients. This leads to introducing decomposable forms as well as modules, orders and maximal orders of a number field. With the use of Minkowski’s geometric methods and his Lemma on Convex Bodies, the Dirichlet’s theorem on the group of units of an order of a number field is proved, and then used to classify, up to similarity, all full modules of that field. The developed general theory is thoroughly discussed in the case of quadratic forms and the related quadratic number fields.
The course next covers in significant detail the topic which led to the development of modern Algebra: the divisibility and unique factorization in number fields. Starting with Euclidean rings, which have unique factorization, the course goes on developing the theory of divisors and valuations, degree of inertia and ramifications of prime divisors. The important Dedekind rings are introduced and studied. This includes divisors and ideals, fractional divisors and the divisor class group of such rings. The results are applied to some special cases of the Fermat’s Last Theorem. Once more the general theory is discussed in detail for quadratic fields.