Let be independent linear forms in . Let .
Prove that is a prime ideal.
let and consider the system of equations from elementary linear algebra we know that this system has either no
solution (which implies that ) or a unique solution (which implies that ) or infinitely many solutions (which implies that for some ). in either case
is clrearly an integral domain and thus will always be a prime ideal.
the infinitely many solutions happens when the number of equations is less than the number of variables, i.e. in this case we can find some variables in terms of others. so wehen you mod
out by you'll again get a polynomial ring over but with the number of variables less than