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.
Where \(p\) is prime.
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.