Glossary

4.2 Properties and calculations of discriminants

3 min read•july 30, 2024

Discriminants are the unsung heroes of algebraic number theory. They measure a field's complexity and play a crucial role in understanding ramification, class groups, and regulators. Think of them as the DNA of number fields, revealing their inner structure and behavior.

Calculating discriminants can be tricky, but there are handy formulas for specific fields. For tougher cases, we've got computational tools and techniques. Discriminants have cool properties too, like being multiplicative and giving us clues about ramification and field structure.

Discriminants of number fields

Definition and fundamental properties

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  • of algebraic number field K measures arithmetic complexity of the field
  • For number field K of degree n over Q, discriminant defined as determinant of n×n matrix [Tr(αiαj)]
    • α1, ..., αn form an integral basis of K
  • Discriminant always non-zero integer and independent of choice of integral basis
  • Absolute value of discriminant relates to volume of fundamental domain of of K
  • Discriminants crucial for understanding ramification in number fields and determining difficulty of computations

Significance in algebraic number theory

  • Used in classification of number fields (particularly in tables of low-degree number fields)
  • directly related to discriminant
    • Provides upper bound for of ideals in ring of integers
  • Crucial for understanding class groups and regulator computations in number fields
  • Determines behavior of primes in extensions of number fields
  • In local field theory, closely related to different exponents and ramification indices
  • Appears in important theorems (, )

Calculating discriminants

Methods for specific number fields

  • Quadratic fields Q(√d) (d integer)
    • Discriminant = 4d if d ≡ 2 or 3 (mod 4)
    • Discriminant = d if d ≡ 1 (mod 4)
  • Q(ζn) (ζn primitive nth root of unity)
    • Formula: ±∏p^(φ(n)(p-2)/(p-1)), p runs over prime divisors of n
    • Computed using trace form or relating to coefficients of defining polynomial
  • Higher degree fields
    • Involve complex techniques (using resultants or norm computations)

Computational tools and techniques

  • Software tools for efficient computation
    • PARI/GP
    • Sage
  • Trace form method
    • Construct matrix of traces and calculate determinant
  • Resultant method
    • Use resultant of minimal polynomial and its derivative
  • Norm computations
    • Utilize norm of different ideal in certain cases

Properties of discriminants

Algebraic properties

  • Multiplicative property
    • For tower of fields L/K/Q: disc(L/Q)=disc(K/Q)[L:K]⋅NormK/Q(disc(L/K))disc(L/Q) = disc(K/Q)^{[L:K]} · Norm_{K/Q}(disc(L/K))
  • Sign determines parity of number of real embeddings of field
  • Prime factors correspond to ramified primes in number field
  • Discriminant of Galois extension K/Q always square integer
  • Magnitude provides information about
    • Density of algebraic integers in field
    • Complexity of arithmetic operations

Ramification and field structure

  • Prime factors of discriminant indicate ramified primes
    • Example: In Q(√-5), discriminant = -20, so 2 and 5 are ramified
  • Unramified primes have simpler splitting behavior
  • Larger absolute value of discriminant generally indicates more complex field structure
    • Example: Q(√-163) has discriminant -163, indicating a simpler structure than Q(√-5)

Discriminants in algebraic number theory

Applications in field theory

  • Classification of number fields
    • Example: Totally real cubic fields classified by discriminant range
  • Minkowski bound calculation
    • Formula: ∣dK∣1/n≤(4π)r2/nn!nn(4π)r1/2n|d_K|^{1/n} \leq (\frac{4}{\pi})^{r_2/n} \frac{n!}{n^n} (\frac{4}{\pi})^{r_1/2n}
      • Where n = [K:Q], r1 = number of real embeddings, r2 = number of pairs of complex embeddings
  • Class group computations
    • Discriminant affects class number formula
  • Regulator calculations
    • Appears in analytic class number formula

Connections to other algebraic structures

  • Relation to different ideal in extensions
    • Different ideal norm equals absolute value of
  • Appearance in zeta functions and L-functions
    • Example: Dedekind zeta function has discriminant in functional equation
  • Role in Galois theory
    • Galois group structure reflected in discriminant properties
  • Connection to modular forms and elliptic curves
    • Discriminant of elliptic curve related to conductor

Key Terms to Review (21)

Absolute Discriminant: The absolute discriminant is a mathematical invariant associated with a number field that provides essential information about the field's structure and properties. It is derived from the discriminant of the field's ring of integers and plays a crucial role in understanding the arithmetic properties, including the ramification of primes and the behavior of integral elements within the field. The absolute discriminant is particularly important when calculating various properties of algebraic extensions and can be used to distinguish between different number fields.
Carl Friedrich Gauss: Carl Friedrich Gauss was a prominent German mathematician and scientist who made significant contributions to various fields, including number theory, statistics, and algebra. His work laid the groundwork for modern number theory, influencing concepts such as unique factorization and the study of integer solutions, while also advancing mathematical techniques that are essential in understanding discriminants and cyclotomic fields.
Class number computations: Class number computations refer to the process of determining the class number of a number field, which measures the failure of unique factorization in its ring of integers. The class number can provide deep insights into the arithmetic properties of the field, especially in relation to its discriminant. Understanding these computations is essential for studying the structure of the ideal class group, which categorizes the ideals in terms of their equivalence classes.
Conductor-discriminant formula: The conductor-discriminant formula connects the discriminant of a number field with its conductor, providing insights into the arithmetic properties of the field. Specifically, it expresses the relationship between these two invariants, where the discriminant measures the complexity of the field's structure, while the conductor indicates how 'large' or 'small' the primes are in relation to the field. This relationship plays a crucial role in understanding ramification and the behavior of primes in algebraic number theory.
Cubic Fields: Cubic fields are number fields of degree three over the rational numbers, $ ext{Q}$. These fields can be represented in the form $ ext{Q}( heta)$ where $ heta$ is a root of a cubic polynomial with rational coefficients. The study of cubic fields involves understanding their structure, properties, and how they relate to algebraic integers and discriminants, which provides insight into their arithmetic and geometric behavior.
Cyclotomic Fields: Cyclotomic fields are number fields generated by adjoining a primitive root of unity, typically denoted as $$\ ext{zeta}_n = e^{2\pi i / n}$$, where $$n$$ is a positive integer. These fields are significant in number theory, particularly in studying the properties of algebraic integers, Galois groups, and class numbers. Cyclotomic fields reveal deep connections between algebra and geometry through their roots of unity, which also impact discriminants and their calculations, field extensions, and ramification behavior.
David Hilbert: David Hilbert was a prominent German mathematician known for his foundational contributions to various fields, including algebra, number theory, and mathematical logic. His work laid the groundwork for modern mathematics and significantly influenced the development of algebraic number theory.
Dedekind-Kummer Theorem: The Dedekind-Kummer Theorem is a fundamental result in algebraic number theory that relates the behavior of prime ideals in a number field to the roots of unity and the properties of the field's discriminant. This theorem establishes a connection between the discriminant of a number field and the ramification of primes in its extension fields, helping to understand how primes factor in these extensions.
Determining the nature of roots: Determining the nature of roots involves analyzing the solutions of a polynomial equation to understand their characteristics, such as whether they are real or complex, distinct or repeated. This understanding is primarily achieved through the use of the discriminant, which provides critical insights into the nature and behavior of the roots without explicitly calculating them. By evaluating the discriminant, one can quickly determine if a quadratic equation has two distinct real roots, one repeated real root, or two complex roots.
Discriminant: The discriminant is a mathematical quantity that provides crucial information about the properties of algebraic equations, particularly polynomials. It helps determine whether a polynomial has distinct roots, repeated roots, or complex roots, which is essential for understanding the structure of number fields and their extensions.
Field Extension: A field extension is a bigger field that contains a smaller field and allows for more solutions to polynomial equations. This concept helps in understanding how different fields relate to each other, especially when it comes to algebraic numbers, algebraic integers, and the properties of polynomials in those fields.
Hasse's Principle: Hasse's Principle is a concept in number theory that states if a certain property holds for a number in local fields (like the completions of the rational numbers at various primes), then it also holds for that number in the global field, specifically the rational numbers. This principle is crucial when analyzing whether certain equations have solutions in the rational numbers based on their behavior in local fields, connecting various properties in algebraic number theory, such as discriminants, integral bases, and quadratic fields.
Ideal Class Group: The ideal class group is a fundamental concept in algebraic number theory that measures the failure of unique factorization in the ring of integers of a number field. It consists of equivalence classes of fractional ideals, where two ideals are considered equivalent if their product with a principal ideal is again a fractional ideal. This group plays a crucial role in understanding the structure of rings of integers and their relationship to number fields, helping to connect various areas such as discriminants, integral bases, and the properties of Dedekind domains.
Jacobian Criterion: The Jacobian Criterion is a method used to determine whether a polynomial equation has a certain number of distinct roots in algebraic geometry. It relates to the properties of discriminants, as it provides criteria based on the Jacobian matrix to check if a polynomial is smooth or singular at a given point, thus connecting it to the calculations and properties of discriminants.
Minkowski Bound: The Minkowski Bound is a critical concept in algebraic number theory that provides a bound for the non-zero ideal classes in a number field. It essentially helps to determine the size of the class group, which consists of the equivalence classes of fractional ideals. The bound can be calculated using the discriminant of the number field and is instrumental in understanding the structure of the ring of integers and its integral basis, as well as facilitating calculations related to class numbers and ideal class groups.
Norm: In algebraic number theory, the norm of an algebraic number is a value that gives important information about its behavior in relation to a field extension. It can be viewed as a multiplicative measure that reflects how the number scales when considered within its minimal field, connecting properties of elements with their corresponding fields and extensions.
Quadratic reciprocity: Quadratic reciprocity is a fundamental theorem in number theory that establishes conditions under which two distinct odd prime numbers can be said to be quadratic residues of each other. It essentially provides a powerful criterion for determining whether a quadratic equation has solutions in modular arithmetic, particularly relating to prime numbers. This theorem not only connects different areas of mathematics but also has historical significance and implications for understanding the structure of numbers.
Relative Discriminant: The relative discriminant is a key concept in algebraic number theory that measures the non-degeneracy of a quadratic form associated with a field extension. It is particularly important when comparing two number fields and helps determine properties like whether the fields are linearly disjoint or if they share common subfields. Understanding the relative discriminant provides insight into the behavior of primes in these extensions and their splitting behavior.
Ring of Integers: The ring of integers is the set of algebraic integers in a number field, which forms a ring under the usual operations of addition and multiplication. This concept is crucial as it provides a framework for studying the properties and behaviors of numbers in various algebraic contexts, particularly when dealing with number fields, discriminants, and integral bases.
Square-free: A number is called square-free if it is not divisible by the square of any prime number. This property is important when studying discriminants and field extensions, as square-free numbers help to characterize the nature of roots and their relationships in algebraic structures. In particular, square-free discriminants can indicate whether the corresponding polynomial has distinct roots, which affects the behavior of field extensions derived from those polynomials.
Trace of a Field Extension: The trace of a field extension is a function that takes an element from an extension field and maps it to the sum of its conjugates over the base field. This concept is crucial in understanding various properties of field extensions, especially when analyzing discriminants and their calculations, which reflect how the roots of polynomials behave under different field conditions.
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