As I type up the third PMATH 442 assignment, I’m also slowly making my way through the notes, making sure all the arguments are correct, and attempting to understand them. I thought I would make a post outlining some of my thoughts on the material, including points of confusion. I invite anyone to leave a comment. Once I can understand everything enough to speak coherently about it (and this week is over), I promise I will give another (hopefully fruitful) Scattered Comments post for PMATH 442 (I know another one for PMATH 753 is long overdue, but things just got a bit busy, which is to be expected since it is currently the middle of the term).
I’ve fixed a few (minor) typos so far, and revised some notation, for example if
is some polynomial and is a map, I prefer to denote by the polynomial , rather than . Although the latter notation does have the advantage that it gives us an explicit name for the “induced map” , I can deal with referring to that map simply as .
One question I had was the following: let be an extension and be such that each is -algebraic. Is it true that ? If , this follows from one of the first theorems we covered about algebraic elements. The result follows by induction if is finite. Is the general case true, though?
Now consider the converse: if then each is -algebraic (equivalently, the extension is algebraic). This is apparently called Zariski’s Lemma, and was used near the beginning of PMATH 764 to prove Hilbert’s Nullstellensatz. We did not prove it in PMATH 442. The converse of the more general statement appears in my notes: if , then is algebraic, which implies only has algebraic elements. Why is this true? Also, if only has algebraic elements, does it follow that is algebraic? I want to know exactly which implications regarding this stuff break down when is infinite.
I’m also not that satisfied with the proof of the Fundamental Theorem of Galois Theory presented today. We had
but how does this show that ? Why is ? I agree that , though — because is Galois — so .
I will probably post more later. Also, my camera is back from repair 😀