Matt

nevermind

algebraic numbers

i was thinking about representing them as some polynomial that they are the root of, but it seems equally as trivial

well just basic things like a | b and b | c implies a | c and such

I have a quick question, i'm asked to prove some basic division facts about algebraic numbers...but i'm going about this the same way i would for integers...should i be doing something else?

its described in the first section of orbits and stabilizers : en.wikipedia.org/wiki/Group_action

quick notational question: for a set X and a group G, we have X^G which is the set of fixed points. Isn't this the same as the stabilizer? Why do we have both kinds of notation? I guess the only thing i can think of is that X^H would be useful if we wanted to look at H, a subgroup of G

@Alex thanks, I know that result. I just couldn't remember where to look

ok well replace Z and Q with different infinitely countable sets

I understand thats the bijection, but if simply know Z < X < Q, nothing more, is there some result that says that X has to be countably infinite if its between two countably infinite sets?

I have a quick question. Lets say we have a set between the integers and the rationals, X. Then Z < X < Q. If you want to show that X is countably infinite, then you can easily find a bijection to Z. But if we have some arbitrary X, then is there a theorem to use?

So then we know that the image of R is a subring of S so the map is injective

if I is the intersection of R and J, is it necessarily true that I = f^-1(J)?

So R/f^-1(J) is isomorphic to the image of R

I have a quick question about the isomorphism theorems for rings. I have R a subring of S and J and ideal in S. then let I = the intersection of R and J. I want to show R/I -> S/J is injective. I've set up a homomorphism R -> S/J and shown the kernel is the preimage of J