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ABOUT THE COURSE:
Algebraic number theory involves the study of global fields (number fields and function fields), their associated local fields obtained via completions (such as the p-adic integers), and the arithmetic information encoded in L-functions. Following the approach of classical references such as Cassels– Fröhlich and Lang's Algebraic Number Theory, we develop the necessary commutative algebra — integral extensions, valuation rings, discrete valuation rings, and Dedekind domains — that is an integral part of this subject. The title of the course reflects this perspective and is a deliberate reference to Eisenbud’s Commutative Algebra with a View Toward Algebraic Geometry.
The two central theorems of a standard algebraic number theory course are the finiteness of class groups and Dirichlet's unit theorem, both of which classically require Minkowski's geometry of numbers. In this 8- week course, we omit Dirichlet's unit theorem and instead present the finiteness of class groups via a more algebraic approach.
PREREQUISITES: Rings and modules, Commutative Algebra (Localization and Noetherian rings), Galois theory
INDUSTRY SUPPORT: Cryptography and coding theory based
Algebraic number theory involves the study of global fields (number fields and function fields), their associated local fields obtained via completions (such as the p-adic integers), and the arithmetic information encoded in L-functions. Following the approach of classical references such as Cassels– Fröhlich and Lang's Algebraic Number Theory, we develop the necessary commutative algebra — integral extensions, valuation rings, discrete valuation rings, and Dedekind domains — that is an integral part of this subject. The title of the course reflects this perspective and is a deliberate reference to Eisenbud’s Commutative Algebra with a View Toward Algebraic Geometry.
The two central theorems of a standard algebraic number theory course are the finiteness of class groups and Dirichlet's unit theorem, both of which classically require Minkowski's geometry of numbers. In this 8- week course, we omit Dirichlet's unit theorem and instead present the finiteness of class groups via a more algebraic approach.
PREREQUISITES: Rings and modules, Commutative Algebra (Localization and Noetherian rings), Galois theory
INDUSTRY SUPPORT: Cryptography and coding theory based
