A field is a ring where the multiplication function has an inverse.
The integers, addition and multiplication form a ring, but not a group.
The rational numbers (except \(0\)), addition and multiplication form a field (and a ring).
The real numbers and complex numbers also form fields.
Finite number of elements.
A bilinear map (or function) is a map from two inputs to an output which preserves addition and scalar multiplication. This is in contrast to a linear map, which only has one input.
In addition, the function is linear in both arguments.
That is if function \(f\) is bilinear then:
That is, if any input is \(0\) in an additative sense, the value of the map must be zero.