In-Depth Notes on Fields and Field Theory

Fields Introduction

  • Definition of Fields: A field is a set F equipped with two closed binary operations (addition and multiplication) that satisfy specific axioms. This generalizes familiar sets like rational numbers (Q) and real numbers (R).

Learning Objectives

  • Define a field and prove its properties.
  • Identify when Zn is a field and understand the implications.

Field Axioms

A field must satisfy the following properties for all x, y, z in F:

  1. Associativity:
    • Addition: x + (y + z) = (x + y) + z
    • Multiplication: x · (y · z) = (x · y) · z
  2. Commutativity:
    • Addition: x + y = y + x
    • Multiplication: x · y = y · x
  3. Identity Elements:
    • There exist unique elements 0F (additive identity) and 1F (multiplicative identity) such that
      • x + 0_F = x
      • x · 1_F = x
  4. Additive Inverses:
    • For each x in F, there exists n in F such that x + n = 0_F.
  5. Multiplicative Inverses:
    • For any non-zero x in F, there exists r in F such that x · r = 1_F.
  6. Distributivity:
    • Multiplication distributes over addition: x · (y + z) = (x · y) + (x · z).
Examples of Fields

  • Fields:

    • Real Numbers (R)
    • Rational Numbers (Q)

  • Non-fields:

    • Integers (Z) - lacks inverses for multiplication.
    • Natural Numbers (N) - lacks additive inverses.

Zn as a Field

  • Sets:
    • Z2 = {0, 1}, addition and multiplication mod 2 is a field.
    • Z3 = {0, 1, 2}, addition and multiplication mod 3 is also a field.
Condition for Zn to be a Field

  • For any integer n >= 2, Zn may not be a field if it doesn't satisfy multiplicative inverses.

  • Example:

    • Z4 = {0, 1, 2, 3} lacks a multiplicative inverse for 2 since 2 · 2 = 0. Hence, it is not a field.
Advanced Example: Polynomials over Z2
  • Let P1(Z2) = { ax + b : a, b in Z2 }.
  • This set satisfies field properties resulting in a field with 4 elements: {0, 1, x, x + 1}.

Uniqueness of Identity and Inverses

  • Additive & Multiplicative Identities are Unique:
    • For x in F, if x + n = 0, then n is unique. Similarly, for multiplicative inverses.
Exercises
  • Prove that the additive and multiplicative identities in a field are unique.
  • If F is a field and x, y ∈ F, then x · y = 0 implies either x = 0 or y = 0.

Theorem 8: Fermat’s Little Theorem

  • If p is prime and p does not divide a, then a^(p-1) ≡ 1 (mod p).
  • Corollary 10: For any a and prime p, a^p ≡ a (mod p).
Zn as a Field if n is Prime
  • Theorem 12: Zn is a field iff n is prime.
    • (If direction) If p is prime, all elements have multiplicative inverses by Fermat's theorem.
    • (Only If direction) If n is not prime (composite), there exist non-zero r, s in Zn such that r · s = 0; thus Zn cannot be a field.
Note on Field Elements
  • Fields may contain ploynomials or other structures such as P1(Z2) which do not have prime number of elements but are still classified as fields.

Exercise Example

  • Find the inverse of [22] in Z41.