Number fields are fascinating extensions of rational numbers, opening up a world of algebraic structures. They're formed by adding algebraic numbers to the rationals, creating finite-dimensional vector spaces over Q with unique properties and structures.

These fields are the backbone of algebraic number theory, offering insights into number systems beyond the rationals. They come with special characteristics like unique factorization of ideals and a ring of integers, making them crucial for understanding deeper algebraic concepts.

Number Fields and Properties

Definition and Characteristics

• Number field represents finite extension of rational numbers Q, typically denoted as K
• Obtained by adjoining algebraic numbers to Q, forming algebraic extensions
• Degree of number field K over Q, denoted [K:Q], measures dimension of K as vector space over Q
• Characterized by minimal polynomial, monic irreducible polynomial over Q with root in the field
• Algebraic closure contains all roots of polynomials with coefficients in that field
• Possess properties of perfect fields and characteristic zero

Key Properties and Structure

• Vector space structure over Q allows for basis representation
• Algebraic elements generate the field over Q
• Possess unique factorization of ideals in their ring of integers
• Admit a unique maximal order, known as the ring of integers
• Support Galois theory, with Galois group describing field automorphisms
• Allow for arithmetic operations (addition, multiplication) closed within the field

Examples of Number Fields

Common Types of Number Fields

• Quadratic fields formed by adjoining square roots to Q (Q(√2), Q(√-1))
  • Q(√2) contains elements of form a + b√2, where a, b are rational
  • Q(√-1) is the field of Gaussian rationals
• Cyclotomic fields created by adjoining roots of unity to Q
  • Q(ζn) where ζn is a primitive nth root of unity
  • Important in number theory and Galois theory
• Cubic fields obtained by adjoining cube roots to Q (Q(∛2))
  • Contains elements of form a + b∛2 + c(∛2)², where a, b, c are rational
• Field of algebraic numbers (Q̄) as union of all finite algebraic extensions of Q

Specialized Number Fields

• Real algebraic number fields (subfields of real numbers)
  • Example: Q(√2), Q(∛2)
• Complex algebraic number fields (contain complex numbers)
  • Example: Q(i), Q(e^(2πi/3))
• Splitting fields of polynomials over Q
  • Example: Splitting field of x³ - 2 over Q is Q(∛2, ω) where ω is a primitive cube root of unity

Number Fields and Extensions

Field Extension Properties

• Every number field K forms field extension of Q, with Q as prime subfield
• Tower law applies: [L:Q] = [L:K][K:Q] for extensions L/K and K/Q
• Primitive element theorem ensures every finite separable extension of a field is simple
  • Applicable to number fields over Q due to characteristic zero
• Galois group Gal(K/Q) describes automorphisms of K fixing Q
  • Crucial in understanding field structure and symmetries

Advanced Extension Concepts

• Normal closure of number field K is smallest normal extension containing K
• Compositum of number fields creates larger fields
  • Degree often product of component field degrees
  • Example: Compositum of Q(√2) and Q(√3) is Q(√2, √3) with degree 4 over Q
• Separable and inseparable extensions (all extensions of Q are separable)
• Algebraic vs transcendental extensions (number fields are algebraic)

Degree and Minimal Polynomial of a Number Field

Degree and Its Significance

• Degree of number field K equals degree of its minimal polynomial over Q
• Determines dimension of K as vector space over Q
• Influences complexity of field arithmetic and structure
• Related to index of subgroup in Galois group

Minimal Polynomial Properties

• Minimal polynomial of algebraic number α is monic irreducible polynomial of least degree with α as root
• For K = Q(α), minimal polynomial of α generates ideal of polynomials in Q[x] with α as root
• Roots of minimal polynomial called conjugates, crucial for field structure
  • Example: For Q(√2), minimal polynomial is x² - 2, conjugates are √2 and -√2
• Discriminant computed using minimal polynomial, provides information about field's arithmetic properties
  • Example: Discriminant of Q(√d) is 4d or d depending on whether d ≡ 2,3 (mod 4) or d ≡ 1 (mod 4)

Tools and Applications

• Eisenstein's criterion and rational root theorem help determine irreducibility of polynomials
  • Useful for identifying minimal polynomials of number fields
• Trace and norm of elements expressible in terms of minimal polynomial coefficients
  • For α with minimal polynomial x^n + a_{n-1}x^(n-1) + ... + a_1x + a_0:
    • Trace(α) = -a_{n-1}
    • Norm(α) = (-1)^n a_0
• Minimal polynomial determines field automorphisms and splitting field