11Sxx Algebraic number theory (local fields)
This subtopic studies algebraic number theory in local fields, with emphasis on valuations, completions, local zeta functions, and ramification.
Specific topics
11S05 Polynomials over local fields
Overview
11S05 studies polynomials over local fields within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Polynomials over local fields (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for polynomials over local fields
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S15 Ramification and extension theory
Overview
11S15 studies ramification and extension theory within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Ramification and extension theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for ramification and extension theory
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S20 Galois theory
Overview
11S20 studies galois theory within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Galois theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for galois theory
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S23 Iwasawa theory
Overview
11S23 studies iwasawa theory within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Iwasawa theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for iwasawa theory
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S25 Galois cohomology
Overview
11S25 studies galois cohomology within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Galois cohomology (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for galois cohomology
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S31 Class field theory
Overview
11S31 studies class field theory within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Class field theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for class field theory
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S37 Langlands-Weil conjectures, nonabelian class field theory
Overview
11S37 studies langlands-weil conjectures, nonabelian class field theory within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Langlands-Weil conjectures, nonabelian class field theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for langlands-weil conjectures, nonabelian class field theory
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S40 Zeta functions and $L$-functions
Overview
11S40 studies zeta functions and l-functions within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Zeta functions and L-functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for zeta functions and l-functions
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S45 Algebras and orders
Overview
11S45 studies algebras and orders within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Algebras and orders (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for algebras and orders
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S70 $K$-theory of local fields
Overview
11S70 studies k-theory of local fields within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
K-theory of local fields (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for k-theory of local fields
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S80 Other analytic theory
Overview
11S80 studies other analytic theory within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Other analytic theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for other analytic theory
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S82 Non-Archimedean dynamical systems
Overview
11S82 studies non-archimedean dynamical systems within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Non-Archimedean dynamical systems (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for non-archimedean dynamical systems
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S85 Other nonanalytic theory
Overview
11S85 studies other nonanalytic theory within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Other nonanalytic theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for other nonanalytic theory
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks
11S90 Prehomogeneous vector spaces
Overview
11S90 studies prehomogeneous vector spaces within algebraic number theory over local fields. It highlights core definitions, key theorem patterns, and methods that connect structural and computational viewpoints.
Related Wikipedia Page
Prehomogeneous vector spaces (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and examples for prehomogeneous vector spaces
- Interactions between algebraic structures, analytic estimates, and effective computation
- How local results and global heuristics inform each other in modern number theory
Typical Uses
Used to select proof frameworks, compare equivalent formulations, and support research-grade computation and verification in number-theoretic work.
Applications
- Foundational progress in arithmetic and field theory
- Algorithm design, complexity analysis, and implementation
- Links to coding theory, logic, and cryptographic methods
References
Recommended Textbooks