1  Fields

1.1 Cartesian products, binary functions, binary operations

A Cartesian product \(A\times B\) is defined to be

\[A\times B=\left\{ \left(a,b\right)|a\in A,b\in B\right\}\]

i.e. it is the set of all ordered pairs of elements. The original example is the real 2D plane \(\mathbb{R}^{2}=\mathbb{R}\times\mathbb{R}\). To see this, note that each point in the plane can be assigned Cartesian co-ordinates \(\left(x,y\right)\). This is an ordered pair, since \(\left(x,y\right)\ne\left(y,x\right)\) (order matters).

A binary function is any function that takes in two arguments. We denote this

\[f:A\times B\rightarrow C\]

meaning \(f\) takes in elements of \(A\) and \(B\) and returns elements in \(C\). For example, \(\alpha\cdot|v\rangle=|v'\rangle\), where \(\alpha\) is a number, \(|v\rangle\) and \(|v'\rangle\) are vectors, and \(\cdot\) is multiplication. We can denote this

\[\cdot:\mathbb{R}\times\mathcal{V}\rightarrow\mathcal{V}.\]

A binary operation is a binary function in which all three sets are the same:

\[f:A\times A\rightarrow A.\]

For example, multiplying together two real numbers gives another real number:

\[\cdot:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}.\]

1.2 Definition of a Field

A field \(\mathcal{F}\) is a non-empty set \(F\) along with the binary operations

\[+:F\times F\rightarrow F\,\,\text{(\emph{field addition})}\]

and

\[\cdot:F\times F\rightarrow F\,\,\text{(\emph{field multiplication})}\]

obeying the following axioms.

\(\forall\alpha,\beta,\gamma\in F:\)

  • [F1a]: \(\alpha+\left(\beta+\gamma\right)=\left(\alpha+\beta\right)+\gamma\) (associativity of +)

  • [F1b]: \(\alpha\cdot\left(\beta\cdot\gamma\right)=\left(\alpha\cdot\beta\right)\cdot\gamma\) (associativity of \(\cdot\))

  • [F2a]: \(\alpha+\beta=\beta+\alpha\) (commutativity of \(+\))

  • [F2b]: \(\alpha\cdot\beta=\beta\cdot\alpha\) (commutativity of \(\cdot\))

  • [F3a]: \(\exists0\in F|\alpha+0=\alpha\) (additive identity)

  • [F3b]: \(\exists1\in F|\alpha\cdot1=\alpha\) (multiplicative identity)

  • [F3c]: \(0\ne1\) (non-triviality – my name!)

  • [F4a]: \(\exists\left(-\alpha\right)\in F|\alpha+\left(-\alpha\right)=0\) (additive inverse)

  • [F4b]: \(\forall\left(\alpha\ne0\right)\exists\alpha^{-1}\in F|\alpha\cdot\alpha^{-1}=1\) (multiplicative inverse)

  • [F5]: \(\alpha\cdot\left(\beta+\gamma\right)=\left(\alpha\cdot\beta\right)+\left(\alpha\cdot\gamma\right)\) (distributivity of \(\cdot\) over \(+\)).

Alternatively, we can say that a field is an ordered pair \(\mathcal{F}=\left(F,\left\{ +,\cdot\right\} \right)\) obeying the axioms above.

1.3 Examples of Fields

Examples include:

  • \(\mathbb{R}\) (the reals)

  • \(\mathbb{Q}\) (the rationals)

  • \(\mathbb{C}\) (complex numbers)