 Bounded-degree coefficient proof of Hilbert basis theorem

ID: bounded-degree-coefficient-proof-of-hilbert-basis-theorem

For an ideal

I

R [x]

, let

J_{n}

contain the coefficients of

x^{n}

in its elements of degree at most

n

. These are ideals of

R

and form an ascending chain, since multiplying by

x

embeds one into the next. If

R

is a Noetherian ring, the chain stabilizes at

N

, and each

J_{n}

for

0 \leq n \leq N

has finitely many generators. Choose corresponding polynomials in

I

. Their union generates

I

: for an element of degree

d

, subtract a combination of the chosen polynomials for

J_{d}

d < N

, or the polynomials for

J_{N}

multiplied by

x^{d - N}

otherwise. This cancels its leading coefficient and lowers its degree. Induction completes the proof without assuming that

R

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