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 (Galois) fields

Finite number of elements.

Integers mod \(p\) field

Characteristic of a field

Algebra on a field

Bilinear maps

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:







Note that:




That is, if any input is \(0\) in an additative sense, the value of the map must be zero.

Algebra on a field