Key Concepts in Essential Number Fields

Essential number fields form the backbone of number theory, starting with rational numbers and extending to various algebraic structures. These fields help us understand polynomial equations, prime distributions, and the connections between local and global properties in arithmetic geometry.

1. Rational numbers (ℚ)

   • The set of numbers that can be expressed as the quotient of two integers, where the denominator is not zero.
   • Fundamental building blocks of number theory and serve as the simplest number field.
   • Closed under addition, subtraction, multiplication, and division (except by zero).

2. Algebraic number fields

   • Extensions of the rational numbers formed by adjoining roots of polynomial equations with rational coefficients.
   • Can be finite or infinite, with finite extensions being particularly important in number theory.
   • Provide a framework for studying solutions to polynomial equations and their properties.

3. Quadratic fields

   • A specific type of algebraic number field generated by the square root of a rational number.
   • Can be expressed in the form ℚ(√d), where d is a square-free integer.
   • Important for understanding the distribution of prime numbers and class numbers.

4. Cyclotomic fields

   • Fields generated by the roots of unity, specifically the solutions to the equation x^n = 1 for some integer n.
   • Denoted as ℚ(ζ_n), where ζ_n is a primitive n-th root of unity.
   • Play a crucial role in number theory, particularly in the study of modular forms and Galois theory.

5. Gaussian integers

   • Complex numbers of the form a + bi, where a and b are integers.
   • Form a ring that extends the integers and allows for unique factorization.
   • Useful in number theory, particularly in the study of prime numbers and Diophantine equations.

6. Eisenstein integers

   • Complex numbers of the form a + b(ω), where a and b are integers and ω = e^(2πi/3) is a primitive cube root of unity.
   • Form a unique factorization domain, similar to Gaussian integers.
   • Important in algebraic number theory and the study of modular forms.

7. p-adic numbers

   • A system of numbers that extends the rational numbers, focusing on the properties of divisibility by a prime p.
   • Defined using a different metric, leading to a unique topology that is useful in number theory.
   • Provide insights into local properties of numbers and are essential in the study of local fields.

8. Function fields

   • Fields of rational functions over a base field, often used in algebraic geometry.
   • Analogous to number fields but involve functions rather than numbers.
   • Important for understanding geometric properties and arithmetic of algebraic varieties.

9. Global fields

   • Includes both number fields and function fields, providing a unifying framework for studying arithmetic properties.
   • Have a well-defined notion of places, allowing for a comprehensive study of local and global properties.
   • Essential in the development of class field theory and the study of Diophantine equations.

10. Local fields

    • Complete fields with respect to a discrete valuation, often arising from p-adic numbers or completions of number fields.
    • Provide a local perspective on global fields, allowing for the study of local properties of solutions to equations.
    • Crucial for understanding the local-global principle in number theory and arithmetic geometry.