---
date: 2022-01-08 03:48
modification date: Monday 24th January 2022 20:20:09
title: Algebraic Number Theory
aliases: ["Algebraic Number Theory", ANT]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [2021 Class Field Theory Notes](Projects/2021%20Class%20Field%20Theory%20Notes.md)
	- [Unsorted/Galois theory](Unsorted/Galois%20theory.md)
	- [analytic class number formula](Unsorted/analytic%20class%20number%20formula.md)


---

# MOC Algebraic Number Theory


# References

## Texts


- Marcus, *Number Fields*

	> Good source of computational problems
- [**Local Fields**](http://link.springer.com/book/10.1007/978-1-4757-5673-9), J.-P. Serre.
- [**Class Field Theory**](http://www.jmilne.org/math/CourseNotes/cft.html), J.S. Milne.  
- Algebraic number theory, J.W.S. Cassels and A. Frohlich. ([**errata**](http://wwwf.imperial.ac.uk/~buzzard/CFerrata.pdf)).  
- [**Multiplicative Number Theory**](http://link.springer.com/book/10.1007%2F978-1-4757-5927-3), H. Davenport.  
- [**Algebraic Number Theory**](http://www.jmilne.org/math/CourseNotes/ant.html), J.S. Milne.  
- [**Algebraic Number Theory**](http://link.springer.com/book/10.1007%2F978-1-4612-0853-2), S. Lang.  
- [**Introduction to Modern Theory**](http://link.springer.com/book/10.1007/3-540-27692-0),Yu. I. Manin and A. A. Panchishkin.  
 - [**Algebraic Number Theory**](http://link.springer.com/book/10.1007/978-3-662-03983-0), J. Neukirch.  

> Good exposition. First 2.5 chapters contains most of the core material. Supplement with problems from Marcus
- [**A Course in Arithmetic**](http://link.springer.com/book/10.1007/978-1-4684-9884-4), J.-P. Serre.  
- Szamuely, *Galois Groups and Fundamental Groups*

## Notes

- See [Definitions](Projects/2022%20Advanced%20Qual%20Projects/Algebraic%20Number%20Theory/020%20Defiinitions.md)
- [this MO question](http://mathoverflow.net/questions/13282/good-algebraic-number-theory-books)
- [attachments/ant.pdf](attachments/ant.pdf)
- [https://www.math.ucla.edu/~sharifi/algnum.pdf](https://www.math.ucla.edu/~sharifi/algnum.pdf)
- [https://wstein.org/books/ant/ant.pdf](https://wstein.org/books/ant/ant.pdf)
- [https://math.berkeley.edu/~apaulin/NumberTheory.pdf](https://math.berkeley.edu/~apaulin/NumberTheory.pdf)
- [https://kskedlaya.org/cft/preface-1.html](https://kskedlaya.org/cft/preface-1.html)
	- [https://math.ucsd.edu/~kkedlaya/math204b/](https://math.ucsd.edu/~kkedlaya/math204b/)
- Lots of advice: http://www.mathcs.emory.edu/~dzb/advice.html

# Topics

- What is [Gauss' lemma](Gauss'%20lemma.md)?
- What is the [Galois correspondence](Unsorted/Galois%20correspondence.md)?
- Number fields and [ring of integers](Unsorted/ring%20of%20integers.md);
- [Splitting, ramification, inertia](Unsorted/ramification%20index.md) of prime ideals under finite extensions
	- [efd theorem](efd%20theorem.md)
- Main statements of [Unsorted/class field theory](Unsorted/class%20field%20theory.md):
	- [Galois cohomology](Galois%20cohomology.md)
- What is the [Hasse–Minkowski theorem](Hasse–Minkowski%20theorem)?
- What is [Artin symbol](Unsorted/Artin%20symbol.md)?
- What is the product formula for global fields?
- The [ideal class group](Unsorted/class%20group.md) is finite.
- The [unit group](Unsorted/unit%20group.md) is finitely generated.
- [Dirichlet's unit theorem](Dirichlet's%20unit%20theorem.md)
- Minkowski’s Lattice Point Theorem
- The [Minkowski bound](Minkowski%20bound.md)
- The [Unsorted/Kronecker-Weber theorem](Unsorted/Kronecker-Weber%20theorem.md)
- [Hilbert 90](Hilbert%2090.md)
- [Kummer theory](Kummer%20theory.md)
- [Krasner's lemma](Krasner's%20lemma)
- [Weak approximation](Unsorted/Weak%20approximation.md)
- [Strong approximation](Strong%20approximation.md)
- [Unsorted/Chebotarev density](Unsorted/Chebotarev%20density.md)

## Analytic NT

- Dirichlet's theorem on primes in APs
- The [Riemann-Zeta function](Unsorted/Riemann%20Zeta.md) 
- [Poisson summation](Poisson%20summation.md)
- The [Fourier transform](Unsorted/fourier%20transform.md)
- [Jacobi theta](Jacobi%20theta.md)
- The [Gamma function](Gamma%20function.md)
- The [analytic class number formula](analytic%20class%20number%20formula.md) 


# Courses

## MIT OCW in ANT

A useful course in #NT/algebraic which seems to motivate a lot of results in #CA

- [x] 1
[Absolute values and discrete valuations (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec1.pdf)

- [x] 2
[Localization and Dedekind domains (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec2.pdf)

- [x] 3
[Properties of Dedekind domains, ideal class groups, factorization of ideals (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec3.pdf)

- [x] 4
[Étale algebras, norm and trace (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec4.pdf)

- [x] 5
[Dedekind extensions (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec5.pdf)

- [x] 6
[Ideal norms and the Dedekind-Kummer theorem (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec6.pdf)

- [x] 7
[Galois extensions, Frobenius elements, and the Artin map (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec7.pdf)

- [x] 8
[Complete fields and valuation rings (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec8.pdf)

- [x] 9
[Local fields and Hensel's lemmas (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec9.pdf)

- [x] 10
[Extensions of complete DVRs (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec10.pdf)

- [x] 11
[Totally ramified extensions and Krasner's lemma (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec11.pdf)

- [x] 12
[The different and the discriminant (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec12.pdf)

- [x] 13
[Global fields and the product formula (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec13.pdf)

- [x] 14
[The geometry of numbers (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec14.pdf)

- [x] 15
[Dirichlet's unit theorem (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec15.pdf)

- [x] 16
[Riemann's zeta function and the prime number theorem (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec16.pdf)

- [x] 17
[The functional equation (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec17.pdf)

- [x] 18
[Dirichlet _L_-functions and primes in arithmetic progressions (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec18.pdf)

- [x] 19
[The analytic class number formula (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec19.pdf)

- [x] 20
[The Kronecker-Weber theorem (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec20.pdf)

- [x] 21
[Class field theory: ray class groups and ray class fields (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec21.pdf)

- [x] 22
[The main theorems of global class field theory (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec22.pdf)

- [x] 23
[Tate cohomology (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec23.pdf)

- [x] 24
[Artin reciprocity in the unramified case (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec24.pdf)

- [x] 25
[The ring of adeles, strong approximation (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec25.pdf)

- [x] 26
[The idele group, profinite groups, infinite Galois theory (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec26.pdf)

- [x] 27
[Local class field theory (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec27.pdf)

- [x] 28
[Global class field theory and the Chebotarev density theorem (PDF)](https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019/lecture-notes/MIT18_785F19_lec28.pdf)

## Kedlaya, Class Field Theory

**Topics by date (with videos, references, notes, and boards):** See also [https://mediaspace.ucsd.edu/playlist/dedicated/1_mudvfogg/](https://mediaspace.ucsd.edu/playlist/dedicated/1_mudvfogg/) for all of the videos at once.

-   Oct 2 (F): Overview of the course ([https://math.ucsd.edu/~kkedlaya/math204a/algebraic_numbers.pdf](https://math.ucsd.edu/~kkedlaya/math204a/algebraic_numbers.pdf).
-   Oct 5 (M): Gaussian and Eisenstein integers ([https://brilliant.org/wiki/gaussian-integers/](https://brilliant.org/wiki/gaussian-integers/).
-   Oct 7 (W): Eisenstein and other quadratic integers ([https://miro.com/app/board/o9J_kkyj6FU=/)](https://miro.com/app/board/o9J_kkyj6FU=/)).
-   Oct 9 (F): Rings of integers in number fields ([https://brilliant.org/wiki/algebraic-number-theory/ > ring-of-integers](https://brilliant.org/wiki/algebraic-number-theory/#ring-of-integers).  
    This lecture was not fully recorded due to a technical issue.
-   Oct 12 (M): Unique factorization of ideals ([https://miro.com/app/board/o9J_kkyj76U=/)](https://miro.com/app/board/o9J_kkyj76U=/)). References: Neukirch I.2, I.3.
-   Oct 14 (W): Discriminant of a basis, proof of unique factorization, fractional ideals ([https://miro.com/app/board/o9J_kkyhUik=/)](https://miro.com/app/board/o9J_kkyhUik=/)). References: Neukirch I.2, I.3.
-   Oct 16 (F): The lattice of a number field ([https://miro.com/app/board/o9J_kkyhUtM=/)](https://miro.com/app/board/o9J_kkyhUtM=/)). References: Neukirch I.5.
-   Oct 19 (M): Minkowski's theorem ([https://miro.com/app/board/o9J_kkyipr0=/)](https://miro.com/app/board/o9J_kkyipr0=/)). References: Neukirch I.4, I.5, I.6.
-   Oct 21 (W): The class number; the multiplicative lattice of a number field ([https://miro.com/app/board/o9J_kkyg1wg=/)](https://miro.com/app/board/o9J_kkyg1wg=/)). References: Neukirch I.5, I.6, I.7.
-   Oct 23 (F): The multiplicative lattice and the units theorem ([https://miro.com/app/board/o9J_kkyg1z0=/)](https://miro.com/app/board/o9J_kkyg1z0=/)). References: Neukirch I.6, I.7.
-   Oct 26 (M): Computational tools for algebraic number theory ([https://miro.com/app/board/o9J_kkyvON0=/)](https://miro.com/app/board/o9J_kkyvON0=/)).
-   Oct 28 (W): Extensions of Dedekind domains ([https://miro.com/app/board/o9J_kkyvOPA=/)](https://miro.com/app/board/o9J_kkyvOPA=/)). References: Neukirch I.8.
-   Oct 30 (F): continuation ([https://miro.com/app/board/o9J_kkyvOJM=/)](https://miro.com/app/board/o9J_kkyvOJM=/)).
-   Nov 2 (M): Cyclotomic fields ([https://miro.com/app/board/o9J_kkyvOK8=/)](https://miro.com/app/board/o9J_kkyvOK8=/)). References: Neukirch I.10, Marcus chapter 2.
-   Nov 4 (W): Galois groups, ramification, and splitting ([https://miro.com/app/board/o9J_kkyvOL8=/)](https://miro.com/app/board/o9J_kkyvOL8=/)). References: Neukirch I.9.
-   Nov 6 (F): continuation ([https://miro.com/app/board/o9J_kkysR9o=/)](https://miro.com/app/board/o9J_kkysR9o=/)).
-   Nov 9 (M): Localization ([https://miro.com/app/board/o9J_kkysO_k=/)](https://miro.com/app/board/o9J_kkysO_k=/)). References: Neukirch I.11.
-   No lecture on Wednesday, November 11.
-   Nov 13 (F): continuation ([https://miro.com/app/board/o9J_kkysOGs=/)](https://miro.com/app/board/o9J_kkysOGs=/)).
-   Nov 16 (M): Different and discriminant ([https://miro.com/app/board/o9J_kkysOAQ=/)](https://miro.com/app/board/o9J_kkysOAQ=/)). References: Neukirch III.2.
-   Nov 18 (W): continuation ([https://miro.com/app/board/o9J_kkysOB8=/)](https://miro.com/app/board/o9J_kkysOB8=/)).
-   Nov 20 (F): Structure of ramification groups ([https://miro.com/app/board/o9J_kkysOD4=/)](https://miro.com/app/board/o9J_kkysOD4=/)). References: Neukirch II.10.
-   Nov 23 (M): _p_-adic numbers ([https://miro.com/app/board/o9J_kkysOMg=/)](https://miro.com/app/board/o9J_kkysOMg=/)). References: Neukirch II.1.
-   Nov 25 (W): _p_-adic absolute value ([https://miro.com/app/board/o9J_kkysOO4=/)](https://miro.com/app/board/o9J_kkysOO4=/)). References: Neukirch II.2, II.4.
-   No lecture on Friday, November 27.
-   Nov 30 (M): Valuations ([https://miro.com/app/board/o9J_kkysOI8=/)](https://miro.com/app/board/o9J_kkysOI8=/)). References: Neukirch II.3.
-   Dec 2 (W): Extensions of valuations ([https://math.ucsd.edu/~kkedlaya/math204a/extension_valuations.pdf](https://math.ucsd.edu/~kkedlaya/math204a/extension_valuations.pdf).
-   Dec 4 (F): Hensel's lemma ([https://miro.com/app/board/o9J_kkysOL4=/)](https://miro.com/app/board/o9J_kkysOL4=/)). References: Neukirch II.4.
-   Dec 7 (M): Newton polygons ([https://miro.com/app/board/o9J_kkysOUQ=/)](https://miro.com/app/board/o9J_kkysOUQ=/)). References: Neukirch II.6.
-   Dec 9 (W): The Kronecker-Weber theorem: preview of Math 204B ([https://kskedlaya.org/cft/](https://kskedlaya.org/cft/), chapter 1.
-   Dec 11 (F): The local Kronecker-Weber theorem ([https://kskedlaya.org/cft/](https://kskedlaya.org/cft/), chapter 1.