Algebraic Number Theory & Class Field Theory

Prerequisite path Preparation & skip guidance · Elementary Number Theory · Abstract Algebra & Galois Theory · Commutative Algebra

Studying Algebraic Number Theory & Class Field Theory

Unique factorization of elements can fail in number fields, while ideals retain a more stable factorization theory. Milne's introductory notes connect rings of integers, discriminants and Dedekind domains to this change of viewpoint. His class-field notes pursue a later question about abelian extensions. Treat that second work as an advanced continuation requiring local and global foundations, with a substantial distance between the two starting points.

What to work on

Compute rings of integers, traces, norms and discriminants in quadratic fields.
Factor rational primes into prime ideals with ramification and residue degrees.
Distinguish factorization of elements from factorization of ideals.
Use ideal classes and a Minkowski bound to organize class-group calculations.

A starting text

Algebraic Number Theory — James S. Milne. Read chapter 2, Rings of Integers, especially norms, traces, discriminants and finding the ring of integers; chapter 3, Dedekind Domains; Factorization, especially ideal factorization, the class group and finding factorizations; then chapter 4, The Finiteness of the Class Number, for lattices and bounds.

Choosing what to read

Find an integral basis in a quadratic field and compute its discriminant, comparing it with the basis you might first guess. Factor a small rational prime into ideals and check the ramification and residue degrees. Distinguish an ideal being principal from an element factoring uniquely. Use a lattice bound to make a class-group calculation finite. Keep localization and valuation examples nearby before beginning a local or global reciprocity argument.

Algebraic Number Theory

James S. Milne

Compute discriminants and ideal factorizations before advanced arithmetic.

How it reads: Number-field course built around integers, ideals and class-number bounds.
What it expects: Field extensions, ideals, localization and module calculations.
What to keep in mind: Class field theory requires additional local, global and cohomological preparation.
Practice: Public chapter exercises and Appendix A Solutions to the Exercises are present. Sampled entries include hints and sketches; complete worked coverage of every exercise was not audited. Chapter 2 exercises, printed page 44, and chapter 3 exercises, page 67. Appendix A begins on printed page 153; entries use chapter-exercise labels such as 2-1.

Complete public PDF.

Edition, reading range and source

Read chapter 2, Rings of Integers, especially norms, traces, discriminants and finding the ring of integers; chapter 3, Dedekind Domains; Factorization, especially ideal factorization, the class group and finding factorizations; then chapter 4, The Finiteness of the Class Number, for lattices and bounds.

Source · Checked 2026-10-16

Class Field Theory

J. S. Milne

Pursue abelian extensions after number-field and local arithmetic foundations.

How it reads: Advanced local and global class-field continuation.
What it expects: Graduate algebraic number theory; check the specific local and cohomological requirements.
What to keep in mind: It should not be started immediately after elementary congruences or only the selected introductory chapters.
Practice: Public solution coverage was not recorded for this item; check its current exercises before planning independent problem work.

Complete 296-page author PDF, version 4.03 dated August 6, 2020.

Edition, reading range and source

Use the notes' prerequisite discussion to choose a local or global route; no fresh chapter range is asserted.

Source · Checked 2026-10-17

Practice

Number-field exercises with author solution sketches. Chapter 2 exercises, printed page 44, and chapter 3 exercises, page 67. Appendix A begins on printed page 153; entries use chapter-exercise labels such as 2-1.. Public chapter exercises and Appendix A Solutions to the Exercises are present. Sampled entries include hints and sketches; complete worked coverage of every exercise was not audited.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

In Z[i], factor the ideals (5) and (2), identify ramification and residue degrees, and justify the factorizations with norms.
A hint
Factor the rational integers as Gaussian integers. Show the quotients by the proposed prime generators are fields, rather than inferring primality solely from a drawing.
Work through the solution
In $\mathbb Z[i]$ , $5=(2+i)(2-i)$ , so $(5)=(2+i)(2-i)$ as a product of principal ideals. The quotient maps $a+bi\mapsto a+3b$ and $a+bi\mapsto a+2b$ to $\mathbb F_5$ have kernels $(2+i)$ and $(2-i)$ respectively. For example, $i\equiv-2$ modulo $(2+i)$ , so reducing any Gaussian integer gives its stated residue; the same argument applies to $(2-i)$ . Hence both ideals are prime of norm $5$ , with residue degree $1$ . They are distinct: the second quotient map sends $2+i$ to $4\ne0$ , so $2+i\notin(2-i)$ . Both exponents are $1$ , so neither is ramified over $5$ . Their norms multiply to $25=N((5))$ .

Since $(1+i)^2=2i$ and $i$ is a unit, $(2)=(1+i)^2$ . The map $a+bi\mapsto a+b$ to $\mathbb F_2$ has kernel $(1+i)$ , so this is a prime ideal of norm $2$ , residue degree $1$ and ramification index $2$ . Its squared norm is $4=N((2))$ . The equalities of elements up to units establish the ideal factorizations; norms provide independent size checks.

If this was the sticking point:
- Abstract Algebra & Galois Theory — Review quotient fields, kernels of ring maps and principal ideals.
- Elementary Number Theory — Review congruences and sums of two squares.

Already comfortable with this? Skip quadratic-field review if you can determine the full ring of integers rather than just an order. Skip ideal-factorization review if you can do the Gaussian example and state the role of Dedekind domains; class-group and local-field study remain separate gaps to assess.

Review the prerequisites · Compare the full subject collection

Where this can lead

Commutative Algebra — Review integrality, localization and Dedekind-domain arguments when ideal calculations stall.
Modular Forms & Arithmetic Geometry — Approach arithmetic applications after complex analysis and the first theory of modular forms.

Reading-source dates

Algebraic Number Theory · 2026-10-16

Reading choices revised 2026-10-17.

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Algebraic Number Theory

PDF

J. S. Milne · Course notes · Intermediate

A standard graduate course with proofs. A natural bridge from fields and Galois theory to arithmetic of number fields.

Preparation: Galois theory; commutative algebra.

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Class Field Theory

PDF

J. S. Milne · Course notes · Advanced

A detailed advanced continuation of algebraic number theory. Read after local and global number-field foundations, not immediately after elementary congruences.

Preparation: Graduate algebraic number theory.

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Additional reading

Search the complete catalogue.

Graduate reading

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Edition and sources

Version: TIFR volume 7; retypeset digital edition (see PDF imprint for publication year) · Checked 2026-10-11

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Edition and sources

Version: TIFR volume 27; retypeset digital edition (see PDF imprint for publication year) · Checked 2026-10-11

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Preparation: Galois theory; algebraic groups; cohomology.

Use: Research preparation in arithmetic groups.

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Edition and sources

Version: TIFR volume 47; retypeset digital edition (see PDF imprint for publication year) · Checked 2026-10-11

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The official lecture index supplies the full course as separate PDFs; this is not a single merged PDF.

Graduate course notes on number fields, local fields and class-field-theoretic methods. The linked collection provides individual lecture PDFs.

Preparation: Abstract algebra; commutative algebra (course corequisite).

Use: Work through a graduate algebraic-number-theory sequence.

Official course & lecture PDFs Source / terms

Edition and sources

Version: Fall 2025 complete course collection · Checked 2026-10-11

Local Fields

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Purchase or library subscription required; previews are not the complete book.

Studies local fields, ramification and local class field theory through cohomological methods. Start after number fields and discrete valuations.

Preparation: Algebraic number theory; Galois theory; group cohomology.

Official page / access details

Edition and sources

Version: English edition, 1979 · Checked 2026-10-11

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Daniel A. Marcus · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

An algebra-first approach to rings of integers and ideal theory with many exercises. Suitable before a more local or adelic treatment.

Preparation: Abstract algebra; field extensions.

Official page / access details

Edition and sources

Version: Second edition, 2018 · Checked 2026-10-11

Number Theory

Helmut Hasse · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A classical arithmetic reference connecting algebraic and local methods. Compare terminology with modern introductory number-field courses.

Preparation: Graduate algebra; algebraic number theory.

Official page / access details

Edition and sources

Version: English edition, 1980; publisher reprint · Checked 2026-10-11

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Kiran S. Kedlaya · Online lecture notes · Graduate · English · Online text

Official complete course/text index and linked reading material opened without login.

Develops reciprocity from abelian extensions of the rationals through cohomology, local fields and adeles.

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Use: Bridge local and global reciprocity.

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Edition and sources

Version: Author-hosted notes; date not stated · Checked 2026-10-13

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Full PDF opened publicly on the author or university host.

Interweaves rings of integers and ideals with Dirichlet series and prime distribution; the author identifies this as a developing draft.

Preparation: Graduate algebra; real and complex analysis.

Use: Connect algebraic and analytic arithmetic.

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Edition and sources

Version: Version of October 7, 2026 · Checked 2026-10-13

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Full PDF opened publicly on the author or university host.

A long course on Dedekind domains, number fields, ramification and local/global arithmetic.

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Use: Build a graduate arithmetic foundation.

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Edition and sources

Version: Author-hosted notes; date not stated · Checked 2026-10-13

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Andrew V. Sutherland · Lecture-course collection · Graduate · English · Online text

Official complete course/text index and linked reading material opened without login.

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Edition and sources

Version: MIT 18.782, Fall 2013 · Checked 2026-10-13

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Full PDF opened publicly on the author or university host.

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Use: A function-field bridge from algebraic number theory to the arithmetic of curves.

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Edition and sources

Version: 2021, version 1.00; Fall 2021 Math 7315 course · Checked 2026-10-13

(Quaternion) Algebras in Number Theory

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Preparation: Abstract algebra, number fields and familiarity with local fields.

Use: Build the algebraic background for quaternionic approaches to arithmetic; use the published chapters as one collection.

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Edition and sources

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Research reading

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Cambridge permission covers the posted first edition. The author expressly asks readers not to redistribute it.

Develops nonarchimedean differential modules and their analytic structure.

Preparation: Nonarchimedean fields; differential equations; algebra.

Use: Research preparation in p-adic differential modules.

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Edition and sources

Version: Authorized first-edition PDF, 2010; second edition 2022 is distinct in access, not counted again · Checked 2026-10-11

Arithmetic Duality Theorems

J. S. Milne · Research monograph · Research · English · PDF

Electronic edition: CC BY-NC-ND 2.5; see the copyright page for precise conditions.

Develops local and global duality in Galois, étale and flat cohomology.

Preparation: Class field theory; Galois cohomology; schemes.

Use: Advanced reference for arithmetic duality.

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Edition and sources

Version: Second edition, July 2006 · Checked 2026-10-11

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Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.

A specialized arithmetic course on higher-degree forms and related local questions. Study field and valuation foundations before its arithmetic constructions.

Preparation: Algebraic number theory; local fields.

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Edition and sources

Version: TIFR Lectures on Mathematics 59; authorized retypeset historical edition — original date on title page · Checked 2026-10-11