For all scalars \(\{a, b, c\} \subseteq F\), it

• commutes, so

  \(ab = ba\),

• associates, so

  \((ab)c = a(bc)\),

• distributes over addition, so

  \(c(a + b) = ca + cb\),

• has the exactly one multiplicative identity: \(1\), so

  \(1a = a\),

• has the exactly one multiplicative inverse: \(1 / a\), so

  \(\frac{1}{a} a = a^{-1} = 1\).