---

Multiplication

Define

… as the binary operation

\[\begin{gather*}
  \forall a, b \in F, \\[1ex]
  ab \in F.
\end{gather*}
\]

where	\(a, b\)	are factors,
	\(ab\)	is the product, and
	\(F\)	is the field with \(a\) and \(b\).

Denote

… by any of the

\[\begin{equation*}
  ab = a \cdot b = a \times b.
\end{equation*}
\]

Constrain

• Multiplicative identity

  … is \(1\), so

  \[\begin{gather*}
    \forall a \in F, \\[1ex]
    1a  = a.
  \end{gather*}
  \]

  More generally,

  \[\begin{gather*}
    \forall a \in F, \> \exists! b \in F : \\[1ex]
    a b = a,
  \end{gather*}
  \]

  where the identity is

  \[\begin{equation*}
    b = 1.
  \end{equation*}
  \]

• Multiplicative inverse

  … is \(a^{-1}\), so

  \[\begin{gather*}
    \forall a \neq 0 \in F, \\[1ex]
    a a^{-1} = 1.
  \end{gather*}
  \]

  More generally,

  \[\begin{gather*}
    \forall a \neq 0 \in F, \> \exists! b \in F : \\[1ex]
    a b = 1,
  \end{gather*}
  \]

  where the inverse is \(b = 1 / a = a^{-1}.\)

• Commutative property

  … holds, so

  \[\begin{gather*}
    \forall a, b \in F, \\[1ex]
    ab = ba.
  \end{gather*}
  \]

• Associative property

  … holds, so

  \[\begin{gather*}
    \forall a, b, c \in F, \\[1ex]
    (ab)c = a(bc).
  \end{gather*}
  \]

• Distributive property

  … holds over addition, so

  \[\begin{gather*}
    a, b, c \in F, \\[1ex]
    c(a + b) = ca + cb
  \end{gather*}
  \]

  which is the one and only

  link between addition and multiplication
  

  in the field.