1.1 Fields and their properties

Fields are the building blocks of modern algebra, providing a framework for understanding numbers and their operations. They extend the concept of number systems, allowing us to work with rational, real, and complex numbers, as well as more abstract structures.

In this section, we'll dive into the definition and properties of fields, explore examples and non-examples, and examine key theorems. Understanding fields is crucial for grasping more advanced concepts in algebra and number theory.

Fields and their Properties

Definition and Essential Axioms

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• A field is a set F together with two binary operations, addition (+) and multiplication (·), that satisfy the following axioms:
  • Closure under addition and multiplication: For all a, b in F, both a + b and a · b are in F
  • Associativity of addition and multiplication: For all a, b, c in F, (a + b) + c = a + (b + c) and (a · b) · c = a · (b · c)
  • Commutativity of addition and multiplication: For all a, b in F, a + b = b + a and a · b = b · a
  • Existence of additive and multiplicative identity elements: There exist elements 0 and 1 in F such that for all a in F, a + 0 = a and a · 1 = a
  • Existence of additive inverses: For each a in F, there exists an element -a in F such that a + (-a) = 0
  • Existence of multiplicative inverses: For each non-zero a in F, there exists an element a^(-1) in F such that a · a^(-1) = 1
  • Distributivity of multiplication over addition: For all a, b, c in F, a · (b + c) = (a · b) + (a · c)

Properties of Identity and Inverse Elements

• The uniqueness of the additive identity: If 0 and 0' are both additive identities, then 0 = 0'
• The uniqueness of the multiplicative identity: If 1 and 1' are both multiplicative identities, then 1 = 1'
• The uniqueness of additive inverses: For each a in F, if -a and -a' are both additive inverses of a, then -a = -a'
• The uniqueness of multiplicative inverses: For each non-zero a in F, if a^(-1) and (a')^(-1) are both multiplicative inverses of a, then a^(-1) = (a')^(-1)
• The multiplicative identity is not equal to the additive identity: 1 ≠ 0
• The additive inverse of the additive identity is itself: -0 = 0
• The multiplicative inverse of the multiplicative identity is itself: 1^(-1) = 1
• For all a in F, a · 0 = 0

Identifying Fields vs Non-fields

Examples of Fields

• Common examples of fields include:
  • The rational numbers (Q) under the usual addition and multiplication operations
  • The real numbers (R) under the usual addition and multiplication operations
  • The complex numbers (C) under the usual addition and multiplication operations
• Finite fields (F_p) exist for any prime number p, where the elements are the integers modulo p under addition and multiplication modulo p

Examples of Non-fields

• The integers (Z) under the usual addition and multiplication operations do not form a field because not every non-zero integer has a multiplicative inverse within Z
  • For example, 2 does not have a multiplicative inverse in Z because there is no integer a such that 2a = 1
• The set of 2x2 matrices with real entries under matrix addition and multiplication does not form a field because not every non-zero matrix has a multiplicative inverse
  • For example, the matrix [[1, 0], [0, 0]] does not have a multiplicative inverse
• The set of continuous functions on the interval [0, 1] under point-wise addition and multiplication does not form a field because not every non-zero function has a multiplicative inverse
  • For example, the function f(x) = x does not have a multiplicative inverse

Arithmetic Operations in Fields

Performing Addition, Subtraction, Multiplication, and Division

• Addition and multiplication in a field must be performed according to the field axioms, ensuring closure, associativity, commutativity, and distributivity
• The additive identity (0) and multiplicative identity (1) elements must be used appropriately in computations
• Additive inverses are used to perform subtraction: a - b = a + (-b)
• Multiplicative inverses (reciprocals) are used to perform division: a ÷ b = a · b^(-1), for non-zero b

Arithmetic in Finite Fields

• In finite fields (F_p), arithmetic operations are performed modulo p
• The multiplicative inverse of an element a is the unique element b such that ab ≡ 1 (mod p)
  • For example, in F_5, the multiplicative inverse of 2 is 3 because 2 · 3 ≡ 1 (mod 5)
• Addition and multiplication tables can be constructed for finite fields to facilitate computations
  • For example, the addition table for F_3 is:

    +	0	1	2
    0	0	1	2
    1	1	2	0
    2	2	0	1

Theorems of Field Properties

Uniqueness of Identity and Inverse Elements

• The uniqueness of the additive identity: If 0 and 0' are both additive identities, then 0 = 0'
  • Proof: Let 0 and 0' be additive identities. Then, 0 = 0 + 0' = 0'
• The uniqueness of the multiplicative identity: If 1 and 1' are both multiplicative identities, then 1 = 1'
  • Proof: Let 1 and 1' be multiplicative identities. Then, 1 = 1 · 1' = 1'
• The uniqueness of additive inverses: For each a in F, if -a and -a' are both additive inverses of a, then -a = -a'
  • Proof: Let -a and -a' be additive inverses of a. Then, -a = -a + 0 = -a + (a + (-a')) = (-a + a) + (-a') = 0 + (-a') = -a'
• The uniqueness of multiplicative inverses: For each non-zero a in F, if a^(-1) and (a')^(-1) are both multiplicative inverses of a, then a^(-1) = (a')^(-1)
  • Proof: Let a^(-1) and (a')^(-1) be multiplicative inverses of a. Then, a^(-1) = a^(-1) · 1 = a^(-1) · (a · (a')^(-1)) = (a^(-1) · a) · (a')^(-1) = 1 · (a')^(-1) = (a')^(-1)

Properties of Zero and One

• The multiplicative identity is not equal to the additive identity: 1 ≠ 0
  • Proof: Assume 1 = 0. Then, for any a in F, a = a · 1 = a · 0 = 0, which contradicts the existence of non-zero elements in F
• The additive inverse of the additive identity is itself: -0 = 0
  • Proof: -0 is the unique element such that 0 + (-0) = 0. Since 0 + 0 = 0, we conclude that -0 = 0
• The multiplicative inverse of the multiplicative identity is itself: 1^(-1) = 1
  • Proof: 1^(-1) is the unique element such that 1 · 1^(-1) = 1. Since 1 · 1 = 1, we conclude that 1^(-1) = 1
• For all a in F, a · 0 = 0
  • Proof: a · 0 = a · (0 + 0) = a · 0 + a · 0. Adding the additive inverse of a · 0 to both sides yields 0 = a · 0