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Guided learning journeys that build knowledge step by step.
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7819 Paths · page 710 / 782
This path provides a systematic graduate-level introduction to the modularity theorem for elliptic curves over the rationals, covering the necessary background in elliptic curves, modular forms, Galois representations, and the proof of Fermat's Last Theorem. It emphasizes the logical dependencies and culminates in the statement and significance of the modularity theorem and its role in FLT.
This graduate-level path develops the theory of elliptic curves through the lens of number theory, culminating in Mordell's theorem. It begins with foundational algebraic geometry and number theory, then builds up to the group law, rational points, and the proof of Mordell's theorem.
This path guides graduate students from complex analysis prerequisites through the theory of modular forms on congruence subgroups, culminating in L-functions and their number-theoretic applications. It emphasizes the conceptual dependencies and connects analytic and arithmetic aspects.
This path guides graduate students through the mathematical foundations needed to understand the Riemann hypothesis, the distribution of nontrivial zeros of the Riemann zeta function, and its profound implications in number theory. It covers complex analysis, analytic number theory, and the prime number theorem, culminating in the statement and significance of the hypothesis.
This graduate-level path provides a rigorous, analytic-number-theoretic introduction to the Riemann zeta function. It begins with the necessary complex analysis and analytic number theory prerequisites, then develops the zeta function's initial definition, its analytic continuation to the complex plane, and its functional equation. The path culminates in an exploration of the zeros and the prime number theorem, with a brief look at generalizations.
A graduate-level path through the analytic proof of Dirichlet's theorem on primes in arithmetic progressions, covering Dirichlet characters, L-functions, and the crucial non-vanishing result, along with key implications and extensions.
A systematic graduate-level path through the theory of Dirichlet characters and Dirichlet L-functions, culminating in Dirichlet's theorem on primes in arithmetic progressions. Covers necessary algebraic, analytic, and complex analytic prerequisites.
A structured graduate-level path covering the theory of ideals in algebraic number fields, from foundational algebraic concepts to ideal factorization, the class group, and the class number. The path emphasizes rigorous prerequisite chains and culminates in the fundamental theorem of ideal arithmetic and finiteness of the class group.
This graduate-level path explores unique factorization in algebraic number fields. It begins with foundational concepts of number fields and integrality, then develops the theory of Dedekind domains and ideal factorization, and culminates in the ideal class group and class number as measures of the failure of unique factorization.
This path systematically develops the theory of algebraic number fields, focusing on their rings of integers, integral bases, and related arithmetic properties. It starts with foundational algebra and number theory, progresses through field extensions and algebraic integers, and culminates in advanced topics such as discriminants, ideal factorization, and the class group.