11Rxx Algebraic number theory (global fields)
This subtopic studies algebraic number theory in global fields, covering ideals, class groups, units, and arithmetic in number fields and function fields.
Specific topics
11R04 Algebraic numbers; rings of algebraic integers
Overview
11R04 addresses algebraic numbers; rings of algebraic integers in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Algebraic numbers; rings of algebraic integers (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for algebraic numbers; rings of algebraic integers
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R06 PV-numbers and generalizations; other special algebraic numbers
Overview
11R06 addresses pv-numbers and generalizations; other special algebraic numbers in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
PV-numbers and generalizations; other special algebraic numbers (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for pv-numbers and generalizations; other special algebraic numbers
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R09 Polynomials over global fields
Overview
11R09 addresses polynomials over global fields in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Polynomials over global fields (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for polynomials over global fields
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R11 Quadratic extensions
Overview
11R11 addresses quadratic extensions in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Quadratic extensions (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for quadratic extensions
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R16 Cubic and quartic extensions
Overview
11R16 addresses cubic and quartic extensions in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Cubic and quartic extensions (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for cubic and quartic extensions
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R18 Cyclotomic extensions
Overview
11R18 addresses cyclotomic extensions in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Cyclotomic extensions (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for cyclotomic extensions
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R20 Other abelian and metabelian extensions
Overview
11R20 addresses other abelian and metabelian extensions in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Other abelian and metabelian extensions (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for other abelian and metabelian extensions
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R23 Iwasawa theory
Overview
11R23 addresses iwasawa theory in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Iwasawa theory (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for iwasawa theory
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R27 Units and factorization
Overview
11R27 addresses units and factorization in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Units and factorization (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for units and factorization
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R29 Class numbers, class groups, discriminants
Overview
11R29 addresses class numbers, class groups, discriminants in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Class numbers, class groups, discriminants (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for class numbers, class groups, discriminants
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R32 Galois theory
Overview
11R32 addresses galois theory in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Galois theory (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for galois theory
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R33 Integral representations related to algebraic numbers
Overview
11R33 addresses integral representations related to algebraic numbers in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Integral representations related to algebraic numbers (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for integral representations related to algebraic numbers
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R34 Galois cohomology
Overview
11R34 addresses galois cohomology in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Galois cohomology (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for galois cohomology
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R37 Class field theory
Overview
11R37 addresses class field theory in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Class field theory (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for class field theory
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R39 Langlands-Weil conjectures, nonabelian class field theory
Overview
11R39 addresses langlands-weil conjectures, nonabelian class field theory in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Langlands-Weil conjectures, nonabelian class field theory (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for langlands-weil conjectures, nonabelian class field theory
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R42 Zeta functions and $L$-functions of number fields
Overview
11R42 addresses zeta functions and l-functions of number fields in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Zeta functions and L-functions of number fields (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for zeta functions and l-functions of number fields
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R44 Distribution of prime ideals
Overview
11R44 addresses distribution of prime ideals in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Distribution of prime ideals (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for distribution of prime ideals
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R45 Density theorems
Overview
11R45 addresses density theorems in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Density theorems (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for density theorems
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R47 Other analytic theory
Overview
11R47 addresses other analytic theory in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Other analytic theory (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for other analytic theory
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R52 Quaternion and other division algebras
Overview
11R52 addresses quaternion and other division algebras in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Quaternion and other division algebras (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for quaternion and other division algebras
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R54 Other algebras and orders
Overview
11R54 addresses other algebras and orders in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Other algebras and orders (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for other algebras and orders
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R56 Adèle rings and groups
Overview
11R56 addresses adele rings and groups in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Adele rings and groups (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for adele rings and groups
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks
11R58 Arithmetic theory of algebraic function fields
Overview
11R58 addresses arithmetic theory of algebraic function fields in algebraic number theory over global fields. It emphasizes rigorous definitions, main theorem families, and analytic or algebraic techniques used in current number-theory practice.
Related Wikipedia Page
Arithmetic theory of algebraic function fields (Wikipedia)
Useful Links
Key Ideas
- Standard formulations and canonical examples for arithmetic theory of algebraic function fields
- Techniques that combine asymptotic estimates with structural arguments
- Connections to adjacent themes in arithmetic, analysis, and computation
Typical Uses
Used to organize advanced proofs, benchmark conjectures against known results, and build pathways from foundational lemmas to research-level statements.
Applications
- Core theoretical development in number theory
- Algorithmic and computational experimentation
- Cross-links to algebraic geometry, automorphic forms, and cryptography
References
Recommended Textbooks