| | |
Summary: Problem Set 4
The following problem is meant to illustrate the idea in the proof of Hilbert's
Nullstellensatz.
Problem 1. Let k be an algebraically closed eld. Let I = (A2
, B2
) k[A, B]. It
is clear that AB I(V (I)) but that AB / I. By Hilbert's Nullstellensatz, some
power of AB is in I (it is clear, in fact, that (AB)2
I but let's approach this
in the spirit of the proof of the Nullstellensatz). Let I = (A2
, B2
, 1 - (AB)Y )
k[A, B][Y ]. Then V (I ) = 0 so 1 I .
a) Find F, G, H k[A, B][Y ] such that 1 = FA2
+ GB2
+ H(1 - (AB)Y ).
b) Plug Y = 1
AB
into the expression 1 = FA2
+ GB2
|
