Field Axioms

Math 110 1 Field Axioms 2015 The Definition of a Field Definition 1.1. A field is a set F with two binary operations on F called addition, denoted +, and multiplication, denoted · , satisfying the following field axioms: FA0 (Closure under Addition) For all x, y ∈ F, the sum x + y is contained in F FA0 (Closure under Multiplication) For all x, y ∈ F, the product x · y is contained in F. FA1 (Commutativity of Addition) For all x, y ∈ F, x + y = y + x. FA2 (Associativity of Addition) For all x, y, x ∈ F, (x + y) + z = x + (y + z). FA3 (Additive Identity) There exists an element 0 ∈ F such that x + 0 = 0 + x = x for all x ∈ F. FA4 (Additive Inverses) For any x ∈ F, there exists y ∈ F such that x + y = y + x = 0. The element y is called the additive inverse of x and written −x. FA5 (Commutativity of Multiplication) For all x, y ∈ F, x · y = y · x. FA6 (Associativity of Multiplication) For all x, y, z ∈ F, (x · y) · z = x · (y · z). FA7 (Multiplicative Identity) There exists an element 1 ∈ F such that x · 1 = 1 · x = x for all x ∈ F. FA8 (Multiplicative Inverses) For any x ∈ F such that x 6= 0, there exists y ∈ F such that x·y = y ·x = 1. The element y is called the multiplicative inverse of x and denoted x−1 or x1 . FA9 (Distributivity of Multiplication over Addition) For all x, y, z ∈ F, x · (y + z) = x · y + x · z. FA10 (Distinct Additive and Multiplicative Identities) 1 6= 0. FA0 is really a consequence of what it means to say that the operations of addition and multiplication are defined on F, so “closure under addition” and “closure under multiplication” are usually not listed as axioms – but we have included them here as a reminder that they must always hold. FA10 excludes one-element sets from being fields. Definition 1.2. A field F is a finite field if F is a finite set. 2 Examples of Fields Which of the sets (with their usual notion of addition and multiplication) are fields? Which are finite fields? • Integers Z • Complex numbers C • Z/pZ for p prime • Rational numbers Q • 3 × 3 matrices M3 (R) with entries in R • Z/nZ for n composite • 3 × 3 invertible matrices GL3 (R) with entries in R • Polynomials Q[x] in x with rational coefficients • Real numbers R • Nonnegative reals R≥0 Page 1 • (Z/nZ)× units modulo n Math 110 3 Field Axioms 2015 Properties of Fields Theorem 3.1. Identity elements are unique. This means: 1. If 0 and 00 both satisfy 0 + x = x + 0 = x and 00 + x = x + 00 = x for all x in F, then 0 = 00 . 2. If 1 and 10 both satisfy x · 1 = 1 · x = x and x · 10 = 10 · x = x for all x in F, then 1 = 10 . Theorem 3.2. Additive and multiplicative inverses are unique. This means for any x in F, 1. If y, y 0 ∈ F satisfy x + y = 0 and x + y 0 = 0, then y = y 0 . 2. If y, y 0 ∈ F satisfy x · y = 1 and x · y 0 = 1, then y = y 0 . This theorem justifies our referring to “the” additive inverse and “the” multiplicative inverse of x, and using the notation −x and x−1 or x1 to refer to a specific uniquely-determined element. Theorem 3.3. If x ∈ F, then −(−x) = x. Theorem 3.4. If x ∈ F and x 6= 0, then (x−1 )−1 = x. Theorem 3.5. Let F be a field, and let a, b, c ∈ F. If a + b = a + c, then b = c. Theorem 3.6. Let F be a field, and let a, b, c ∈ F. If a · b = a · c and a 6= 0, then b = c. This is a crucial property of fields: we can divide by (or cancel) any nonzero element. Theorem 3.7. Let F be a field. If a ∈ F, then a · 0 = 0. Theorem 3.8. Let F be a field, and let a, b ∈ F. If a · b = 0, then a = 0 or b = 0. In particular, fields have no (nonzero) zero divisors. Page 2