Let be a field. Consider the ring of formal power series in i.e. if , then where .
Multiplication is defined by .
(a) Prove that is a unit if and only if the constant term . (ex. is the inverse of )
(b) Prove that is a Euclidean domain with respect to the norm if is the
first term of that is non-zero.
(c) In the polynomial ring , prove that is irreducible.
(a): let it's obvious that if then there exists no such that because the constant term of is 0.
if then is invertible in define and inductively let and see that
(b) straightforward.
(c) this part is nice! suppose for some then and
hence, without loss of generality, we may assume that for some invertible element to complete the proof of
(c), use induction to show that, in we have for all which is the contradiction we need.
Hebrew University in Jerusalem. Indeed, we studied formal powers series in some algebra course in the second half part of undergraduate studies, and got deep into it in graduate courses. I wrote "at least" since imo there could be some school somewhere where this stuff is studied in 2nd-3rd year.
Now basic ring theory we studied as a final chapter to linear algebra 1 and then again, more thorough, as part of Agebraic Structures in 2nd year, as preparation to Fields and Galois theory, so it could be they're going to teach you guys all this stuff in your last year...but it looks a little too little to me.
My school though is considered as rather strong in algebra in general, with some emphasis in group theory, and perhaps this has something to do.
Tonio
well, it means "it's straightforward but i'm too lazy to write it down!" haha ... anyway, you need to check two conditions.
1) for all : well, from the definition of it's clear that so this part is done.
2) suppose that where we want to prove that there exist such that and either or :
if then choose if then put where now is invertible since
so there exists such that thus so in this case we can choose