proof of Hilbert basis theorem
Let be a noetherian ring and let with . Thencall the initial coefficient
of .
Let be an ideal in . We will show is finitelygenerated, so that is noetherian. Now let be apolynomial
of least degree in , and if have been chosen then choose from of minimal degree. Continuing inductivelygives a sequence of elements of .
Let be the initial coefficient of , andconsider the ideal of initialcoefficients. Since is noetherian, forsome .
Then . For if not then, and for some . Let where .
Then , and and. But this contradictsminimality of .
Hence, is noetherian.