PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Number Theory → Algebraic Number Theory ↓


Ideals


In algebraic number theory, the concept of an ideal is a fundamental building block. Ideals allow mathematicians to generalize arithmetical properties and address questions about divisibility in more complex number systems than the real numbers. This explanation will take a closer look at what ideals are, how they function, and their importance in the broader context of number theory.

What is ideal?

An ideal is a special subset of a ring, which is itself a mathematical structure that generalizes the concepts of addition and multiplication. Let's start by considering a basic example from ring theory, the group of integers ℤ.

Definition of the norm

Formally, given a ring R, an ideal I is a subset of R that satisfies two main properties:

  1. Additive closure: if a and b are elements in I, then a + b is also in I
  2. Absorption Property: If r is any element in R and a is an element in I, then the product r * a is in I

These properties ensure that ideals behave somewhat like 'numbers', closed under addition and multiplication by elements of a larger ring.

Examples of ideals

Let us look at some examples to understand the concept of ideals better:

Example 1: Ideal in the ring of integers

Consider the ring of integers ℤ. A simple and important example of an ideal in ℤ is the set of even numbers. It can be represented as 2ℤ, which means all numbers can be expressed in the form 2k where k is an integer.

2ℤ = {..., -4, -2, 0, 2, 4, 6, ...}

This is actually ideal because:

  • It is closed under addition (e.g., 2 + 4 = 6 is also even).
  • It absorbs multiplication by any integer from ℤ (for example, 3*2 = 6 is even).

Example 2: Principal and non-principal ideals

A prime ideal is generated by a single element from a ring. For the ring ℤ, the ideal (5) generated by 5 consists of all integer multiples of 5, denoted as 5ℤ.

5ℤ = {..., -10, -5, 0, 5, 10, ...}

Non-prime ideals are more complicated and cannot be generated by a single element of the ring.

Operations on ideals

Ideals can be manipulated in various ways, much like numbers, allowing the algebraic structure to be explored further.

Sum of ideals

If I and J are ideals in a ring R, then the sum I + J is defined as:

I + J = {a + b | a ∈ I, b ∈ J}

This operation generates another ideal consisting of all elements that can be expressed as the sum of an element from I and an element from J

Product of ideals

The product of two ideals IJ is a set given by:

IJ = {∑ a_ib_i | a_i ∈ I, b_i ∈ J}

Here, the sum operates on a finite sequence where each term contains elements of I and J. The product of ideals is again an ideal.

Visual understanding of motifs

Let us try to understand the concept of ideals through the following example:

Ideal I. Ideal J

The blue and orange rectangles represent two different ideals, I and J, showing their possible overlapping and additive properties when considering their sum I + J

The importance of ideals

In algebraic number theory, ideals play an important role. They are used to deal with divisibility properties in rings that do not have unique factorization. Ideals help generalize the concept of prime, leading to the definition of prime ideals.

Example: Ideal factorization

In some number fields such as the ring of integers, unique factorization into elements may fail. However, using ideals, we can obtain a form of unique factorization.

6 = 2 * 3

In some rings, 6 may factor as (1 + √-5)(1 - √-5), which shows the failure of unique factorization by elements. However, using ideals, we can express a unique factorization involving prime ideals.

Conclusion

Ideals extend the notion of numbers and divisibility to more complex algebraic structures. By studying ideals, mathematicians can explore the rich arithmetical landscape within rings and number fields, providing deep insight into structures beyond the fundamental set of integers.


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Ideals

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