proof of Hilbert basis theorem
Let be an ideal in . We will show is finitely generated, so that is noetherian. Now let be a polynomial of least degree in , and if have been chosen then choose from of minimal degree. Continuing inductively gives a sequence of elements of .
Let be the initial coefficient of , and consider the ideal of initial coefficients. Since is noetherian, for some .
Then . For if not then , and for some . Let where .
Then , and and . But this contradicts minimality of .
Hence, is noetherian.
|Title||proof of Hilbert basis theorem|
|Date of creation||2013-03-22 12:59:27|
|Last modified on||2013-03-22 12:59:27|
|Last modified by||bwebste (988)|