In today's blog, I will talk about the assumptions behind the idea of a field. I use the concept of a field in the proof for the Division Algorithm for Polynomials.
Definition 1: Ring
A ring R is a nonempty set with two binary operations: addition (a + b) and multiplication (ab).
It has the following properties:
1. Commutative Rule for Addition: a + b = b + a
2. Associative Rule for Addition: (a + b) + c = a + (b + c)
3. Additive Identity: there exists a value 0 ∈ R such that a + 0 = a for all a ∈ R
4. Additive Inverse: for all elements a ∈ R, there exists a value -a ∈ R such that a + -a = 0.
5. Associative Rule for Multiplication: a(bc) = (ab)c
6. Distributive Rule: a(b + c) = ab + ac and (b + c)a = ba + ca
Definition 2: Commutative Ring
A ring that has the following additional property:
7. Commutative Rule for Multiplication: ab = ba
Definition 3: Field
A field is a commutative ring R with unity in which every nonzero element is a unit.
It has all the properties of a commutative ring plus:
8. Multiplicative Identity (unity): there exists a value 1 ∈ R such that such that a * 1 = a for all a ∈ R
9. Multiplicative Inverse (every nonzero element is a unit): for all nonzero elements a ∈ R, there exists an inverse a-1 ∈ R such that a*a-1 = 1.
1. The set of integers Z is a commutative ring with unity 1. [See here for more information on the integers]
2. The set Zn is a commutative ring with unity 1. [See here for more information on modular arithmetic]
3. The set of 2x2 matrices with integer entries is a noncommutative ring with unity. [See here for more information on 2x2 matrices]
4. If p is prime, then the set Zp is a field. [See here for more information on modular arithmetic]
5. The set of rational numbers Q is a field.
6. The set of real numbers R is a field.
7. The set of complex numbers is a field. [See here for more information on complex numbers]
8. The set of Gaussian Integers is a ring. [See here for more information on Gaussian Integers]
9. The set of Eisenstein Integers is a ring. [See here for more information on Eisenstein Integers]