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Mathematics LibreTexts

B: Fields


A \(\textit{field}\) \(\mathbb{F}\) is a set with two operations \(+\) and \(\cdot\), such that for all \(a, b, c \epsilon \mathbb{F}\) 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 \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
  • M2.    There exists a multiplicative identity \(1\).
  • M3.    Multiplication is commutative \(a \cdot b = b \cdot a\).
  • M4.    There exists a multiplicative inverse \(a^{-1}\) if \(a \neq 0\).
  • D.      The distributive law holds \(a \cdot (b + c) = ab + ac\).



Roughly, all of the above mean that you have notions of \(+, -, \times, \div\) just as for regular real numbers.


Fields are a very beautiful structure; some examples are rational numbers \(\mathbb{Q}\), real numbers \(\mathbb{R}\), and complex numbers \(\mathbb{C}\). These examples are infinite, however this does not necessarily have to be the case. The smallest example of a field has just two elements, \(\mathbb{Z}_2 = {0, 1}\) or \(\textit{bits}\). The rules for addition and multiplication are the usual ones save that $$1 + 1 = 0.$$