Fields of Scalars¶

A field consists of a set, \(F\), whose elements are called scalars, together with two algebraic operations: addition and multiplication, for combining every pair of scalars, \(x, y \in F\) to form the new scalars \((x+y) \in F\) and \((x \times y) \in F\).

These operations must satisfy the field axioms:

Axiom 5 (Associativity)

For \(x,y,z \in F\),

\[\begin{split}\begin{aligned} (x+y)+ z &= x+(y+z) \\ (x\times y)\times z &= x \times (y \times z) \end{aligned}\end{split}\]

Axiom 6 (Distributivity)

\[\begin{split}\begin{aligned} (x+y) \times z &= x \times z + y \times z \\ z \times (x + y) &= z \times x + z \times y \end{aligned}\end{split}\]

Axiom 7 (Commutativity)

\[\begin{split}\begin{aligned} x + y &= y + x \\ x \times y &= y \times x \end{aligned}\end{split}\]

Axiom 8 (Zero and Unity)

There are unique and distinct elements, \(0, 1 \in F\), such that

\[\begin{split}\begin{aligned} x+0 &= x = 0 + x \\ x \times 1 &= x = 1 \times x \end{aligned}\end{split}\]

Axiom 9 (Existence of Additive and multiplicative inverses)

For \(x \in F\) there exists a unique element \(-x \in F\), for which

\[x + (-x) = 0 = (-x)+x\]

For each non-zero \(y \in F\) there is a unique element, \(y^{-1} \in F\), the multiplicative inverse, for which

\[y \times (y^{-1}) = 1 = (y^{-1})\times y\]