Complete Algebraic Number Theory Cheat Sheet: From Fundamentals to Applications

Introduction to Algebraic Number Theory

Algebraic Number Theory (ANT) studies algebraic structures related to algebraic numbers – numbers that are roots of non-zero polynomials with rational coefficients. It extends classical number theory by incorporating techniques from abstract algebra, especially field and ring theory. ANT is crucial for solving classical number-theoretic problems like Fermat’s Last Theorem and has applications in cryptography, coding theory, and various areas of pure mathematics.

Core Concepts and Foundations

Algebraic Numbers and Integers

• Algebraic number: A complex number that is a root of a non-zero polynomial with rational coefficients
• Algebraic integer: An algebraic number that is a root of a monic polynomial with integer coefficients
• Minimal polynomial: The irreducible polynomial of lowest degree having the number as a root
• Degree: The degree of the minimal polynomial of an algebraic number

Number Fields

• Number field: A finite extension K of the rational numbers ℚ
• Degree of a field: [K:ℚ], the dimension of K as a vector space over ℚ
• Quadratic field: A number field of degree 2 (e.g., ℚ(√d) where d is a square-free integer)
• Cyclotomic field: ℚ(ζₙ) where ζₙ is a primitive nth root of unity

Rings of Integers

• Ring of integers O_K: The set of all algebraic integers in a number field K
• Integral basis: A basis {ω₁,…,ωₙ} such that O_K = ℤω₁ + … + ℤωₙ
• Discriminant: Measures the “size” of the ring of integers; d_K = det(Tr(ωᵢωⱼ))²

Ideals and Factorization

Ideal Theory

• Ideal: An additive subgroup I of a ring R such that r·i ∈ I for all r ∈ R, i ∈ I
• Principal ideal: An ideal generated by a single element: (a) = aR
• Prime ideal: An ideal P such that if ab ∈ P, then either a ∈ P or b ∈ P
• Maximal ideal: An ideal M ≠ R such that if M ⊂ I ⊂ R, then either I = M or I = R

Factorization of Ideals

• Fundamental theorem: In the ring of integers of a number field, every non-zero ideal factors uniquely as a product of prime ideals
• Unique factorization domains (UFDs): Rings where every non-zero element factors uniquely into irreducibles
• Non-UFDs: Rings like ℤ[√-5] where unique factorization fails

Fractional Ideals

• Definition: A fractional ideal is a set I such that dI is an ideal for some non-zero d
• Inverse: For a fractional ideal I, I⁻¹ = {x ∈ K | xI ⊆ O_K}
• Ideal class group: The group of fractional ideals modulo principal ideals

Class Field Theory

Class Groups and Class Numbers

• Class group: Cl(K) = fractional ideals modulo principal fractional ideals
• Class number: h(K) = |Cl(K)|, the order of the class group
• Hilbert class field: The maximal unramified abelian extension of K

Ramification Theory

• Ramified prime: A prime p is ramified in K if the ideal (p) is divisible by the square of a prime ideal in O_K
• Split prime: A prime p splits in K if (p) factors into distinct prime ideals in O_K
• Inert prime: A prime p is inert in K if (p) remains prime in O_K

Valuations and Completions

p-adic Valuations

• p-adic valuation: v_p(n) = highest power of p dividing n
• p-adic absolute value: |x|_p = p^-v_p(x)
• p-adic numbers: ℚ_p, the completion of ℚ with respect to |·|_p

Local-Global Principle

• Hasse principle: Certain equations have a solution over ℚ if and only if they have a solution over ℝ and over ℚ_p for all primes p
• Exceptions: Examples where the Hasse principle fails, like certain genus 1 curves

Common Methods and Techniques

Calculating the Ring of Integers

1. Find a primitive element α for the field K/ℚ
2. Compute the minimal polynomial f(x) of α
3. Calculate the discriminant Δ of f
4. For each prime p dividing Δ, check for p-power denominators in the basis elements
5. Find an integral basis using the round-2 method or similar techniques

Computing the Class Group

1. Find a bound B such that the class group is generated by prime ideals of norm ≤ B
2. Determine the relations between these generators
3. Apply linear algebra to find the structure of the class group

Solving Diophantine Equations

1. Reduce to an equation in the ring of integers
2. Factor into ideals
3. Convert to an equation of principal ideals
4. Solve unit equations as necessary

Comparison of Number Fields

Common Challenges and Solutions

Challenge: Non-unique Factorization

• Problem: In rings like ℤ[√-5], elements may factor in multiple ways
• Solution: Work with ideals instead of elements, as ideals have unique factorization

Challenge: Computing Class Numbers

• Problem: Class number computation becomes difficult as field discriminant grows
• Solution: Use analytic methods such as L-functions and the class number formula

Challenge: Finding Units

• Problem: The structure of the unit group can be complex
• Solution: Apply Dirichlet’s unit theorem and compute fundamental units using continued fractions

Best Practices and Tips

• Working with Ideals:
  • Use the norm of an ideal to simplify calculations: N(I) = |O_K/I|
  • For quadratic fields, ideals can be represented in standard form (a, b+c√d)
• Computational Methods:
  • Use computer algebra systems like PARI/GP, Magma, or Sage for complex calculations
  • Implement the LLL algorithm for finding short vectors in lattices
• Proving Results:
  • Exploit the connection between ANT and algebraic geometry
  • Use reciprocity laws to understand splitting of primes
  • Apply techniques from Galois theory when working with normal extensions

Tools and Software

• PARI/GP: Excellent for computations in algebraic number fields
• Sage: Open-source mathematical software with extensive number theory functionality
• Magma: Commercial system with powerful algorithms for class group and unit group computations
• KANT/KASH: Specialized for algebraic number theory computations

Further Learning Resources

Textbooks

• “Algebraic Number Theory” by J. Neukirch
• “Algebraic Number Theory” by A. Fröhlich and M. Taylor
• “A Classical Introduction to Modern Number Theory” by K. Ireland and M. Rosen

Online Resources

• LMFDB (L-functions and Modular Forms Database): https://www.lmfdb.org/
• Keith Conrad’s expository papers: https://kconrad.math.uconn.edu/blurbs/
• Number Theory Web: http://www.numbertheory.org/

Advanced Topics for Further Study

• Iwasawa Theory
• Modular Forms and L-functions
• Elliptic Curves and Modular Forms
• Arithmetic of Higher-dimensional Varieties

This cheat sheet covers the fundamental aspects of algebraic number theory while providing practical methods and techniques used by researchers and practitioners in the field. The hierarchical structure makes it easy to locate specific information, whether you’re a beginner or have intermediate knowledge of the subject.