Page 317 - 35Linear Algebra

B

Fields

Definition A field F is a set with two operations + and ·, such that for all
a, b, c ∈ F the following axioms are satisfied:

A1. Addition is associative (a + b) + c = a + (b + c).

A2. There exists an additive identity 0.

A3. Addition is commutative a + b = b + a.

A4. There exists an additive inverse −a.

M1. Multiplication is associative (a · b) · c = a · (b · c).

M2. There exists a multiplicative identity 1.

M3. Multiplication is commutative a · b = b · a.

M4. There exists a multiplicative inverse a −1 if a 6= 0.

D. The distributive law holds a · (b + c) = ab + ac.

Roughly, all of the above mean that you have notions of +, −, × and ÷ just
as for regular real numbers.
Fields are a very beautiful structure; some examples are rational num-
bers Q, real numbers R, and complex numbers C. These examples are in-
finite, however this does not necessarily have to be the case. The smallest

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