[wbl]

I was baffled enough by this comment to take my copy and look. The Nullstellensatz is an exercise late in the book long after Noetherian rings are introduced and they don't even do the Rabinowitz trick in the hints as they have enough theory to hit it the hard way. Determinants are nowhere to be found.

[Garlef]

It's been a few years (~ 15) since I read it.

The determinant trick I'm referring to is used in prop 2.4 (in the Version I found via Google).

They use the determinant of the adjugate matrix if I remember correctly... But they just hit the reader with this without any motivation or even naming it.

[wbl]

For Nakayama lemma (but they give a cleaner proof later). Note that the book was from the 1960's and from lectures at Oxford: matrix manipulations like that were a lot more commonplace.