# Fields

## Fields

### Fields

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.

## 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:

$$X=aM+bN$$

$$Y=cO+dP$$

$$f(X,Y)=f(aM+bN,cO+dP)$$

$$f(X,Y)=f(aM,cO+dP)+f(bN,cO+dP)$$

$$f(X,Y)=f(aM,cO)+f(aM,dP)+f(bN,cO)+f(bN,dP)$$

$$f(X,Y)=acf(M,O)+adf(M,P)+bcf(N,O)+bdf(N,P)$$

Note that:

$$f(X,Y)=f(X+0,Y)$$

$$f(X,Y)=f(X,Y)+f(0,Y)$$

$$(0,Y)=0$$

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