[Finite-Dimensional Linear Algebra] 2.1 Fields

Definition of a field

Let `F` be a nonempty set with two defined operations for `\alpha , \beta \in F` called addition and multiplication.

• addition: `\alpha + \beta \in F`
• multiplication: `\alpha\beta \in F`

 

`F` is a field iff (if and only if) the operations satisfy the following properties: 

A. Axioms for Addition

1. Associative property of addition
   • for all `\alpha , \beta , \gamma \in F`, ``(\alpha + \beta) + \gamma = \alpha + (\beta + \gamma)``
2. Commutative property of addition
   • for all `\alpha , \beta \in F`, ``\alpha + \beta = \beta + \alpha``
3. Existence of an additive identity
   • there exists a unique element 0 of `F` for all `\alpha \in F` such that ``\alpha + 0 = \alpha``
4. Existence of additive inverses
   • for each `\alpha \in F`, there exists a unique element `-\alpha \in F` such that ``\alpha + (-\alpha ) = 0``

B. Axioms for Multiplication

1. Associative property of multiplication
   • for all `\alpha , \beta , \gamma \in F`, ``(\alpha \beta )\gamma = \alpha (\beta \gamma)``
2. Commutative property of multiplication
   • for all `\alpha , \beta \in F`, ``\alpha \beta = \beta \alpha``
3. Existence of a multiplicative identity
   • there exists a unique nonzero element 1 of `F` for all `\alpha \in F` such that ``\alpha \cdot 1 = \alpha``
4. Existence of multiplicative inverses
   • for each `\alpha \in F`, `\alpha \neq 0` there exists a unique element `\alpha^{-1} \in F` with the property that ``\alpha \alpha^{-1} = 1``

C. Distributive Axiom

1. Distributive property of multiplication over addition
   • for all `\alpha , \beta , \gamma \in F`, ``\alpha (\beta + \gamma ) = \alpha \beta + \alpha \gamma``

Infinite Fields

`(\mathbb{Q} , +, \times ) \leq ( \mathbb{R} , +, \times ) \leq ( \mathbb{C} , +, \times )`

• `\mathbb{Q}  =  \left\{ \frac{n}{m} \mid m, n \in Z, m \neq 0 \right\}`, the set of all rational numbers
• `\mathbb{R}` = the set of all real numbers
• `\mathbb{C}  =  \left\{ a + bi \mid a, b \in R \right\}`, the set of all complex numbers

 

Q. Why are `( \mathbb{Z} , +, \times )` and `( \mathbb{N} , +, \times )` NOT fields? 

• `\mathbb{Z}  = {..., -2, -1, 0, 1, 2, ...}`, the set of all integers
• `\mathbb{N}  = {1, 2, 3, ...}`, the set of all natural numbers 

A. The set `\mathbb{Z}` of integers violates axiom (B4), the existence of multiplicative inverses; only integers 1 and -1 have multiplicative inverses that are also integers.

For each `n \in \mathbb{Z}, n \geq 2` or `n \leq -2`,

``\left| n \right| > 1``

``\Rightarrow 0 < \left| \frac{1}{n} \right| < 1``

``\Rightarrow \frac{1}{n} \notin \mathbb{Z}``

The set of natural numbers `\mathbb{N} \subset \mathbb{Z}` is also not a field. It violates axioms (A3), (A4), and (B4). 

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Basic Properties of Fields

Theorem (Cancellation Properties) Let `\alpha , \beta , \gamma` be elements of field `F`. Then,

1. if `\alpha , \beta , \gamma \in F` and `\alpha + \gamma = \beta + \gamma`, then `\alpha = \beta`
2. if `\alpha , \beta , \gamma \in F, \alpha \gamma = \beta \gamma`, and `\gamma \neq 0`, then `\alpha = \beta`

 

PROOF (1)

Given the following statement ``\alpha + \gamma = \beta + \gamma ,`` Add the additive inverse of `\gamma` on both sides: ``(\alpha + \gamma ) + (-\gamma ) = (\beta + \gamma ) + (-\gamma )`` By associative property, ``\alpha + (\gamma  + (-\gamma )) = \beta + (\gamma  + (-\gamma ))`` Since we know that `\gamma + (-\gamma ) = 0`, ``\alpha + 0 = \beta + 0`` ``\Rightarrow \alpha = \beta``

 

Theorem Let `F` be a field. Then, for each `\alpha \in F`,

1. `0 \cdot \alpha = 0`
2. `-1 \cdot \alpha = -\alpha`

 

PROOF (1)

As stated by axiom (A3),  ``0 + \alpha = \alpha`` Multiply both sides by `\alpha`: ``(0 + \alpha )\alpha = \alpha \cdot \alpha`` The distributive property yields ``0 \cdot \alpha + \alpha \cdot \alpha = \alpha \cdot \alpha`` By cancellation property, ``0 \cdot \alpha = 0``

 

PROOF (2)

Assume the following statement: ``\alpha + (-1)\cdot \alpha = 0`` Since `\alpha = 1 \cdot \alpha`, ``1 \cdot \alpha + (-1) \cdot \alpha = 0`` By distributive property, ``(1 + (-1))\alpha = 0`` ``\Rightarrow 0 \cdot \alpha = 0`` Hence the statement is true and `-1 \cdot \alpha` is a unique additive inverse of `\alpha`, ``-1 \cdot \alpha = -\alpha``