Recent content by Theorem.

From my experiences, I wish I had taken Topology either as a prerequisite to analysis or as a co-requisite. Although it is perfectly possible to do well in analysis without topology, it is a beautiful subject that makes many of the arguments in analysis a lot clearer- as many arguments in...

Okay I just got an idea. We can suppose towards a contradiction that \alpha \notin \mathbb{Q}[\alpha^3] Then the polynomial x^3-\alpha^3 is irreducible in \mathbb{Q}[\alpha^3] since \alpha is real by assumption and so \mathbb{Q}[\alpha^3] is contained in \mathbb{R} and the other two roots of...

Homework Statement Show that if \alpha is real and has degree 10 over \mathbb{Q} then \mathbb{Q}[\alpha]=\mathbb{Q}[\alpha^3] Homework Equations The Attempt at a Solution It is clear that \mathbb{Q}[\alpha^3]\subset \mathbb{Q}[\alpha]. This gives us the sequence of fields...

I would definitely focus more on the mathematics -you still have the option for something like a quant, but you really aren't restricted to that, there are many industrial jobs in mathematics now, and there is also the academic route of course. I can't speak for an econ major

There isn't enough information up right now to make assumptions about what the school will really offer. It could be a great thing as an extra-curricular for students who are really interested in mathematics, and could help them get the math education that north america doesn't really offer...

It is usually necessary for at least a chapter of the course when you deal with systems, but I know that in my linear algebra class a lot of the students didn't have linear algebra- they just had to spend more time learning the linear algebra. The instructor also did a short review of some of...

I don't necessarily agree with this last student. Even if for some reason it WAS just a reading course- It would be graduate level reading providing the same advantage as a graduate course. All the students I know in summer research actually do research. Sure they have to spend about half the...

Yes you still have lots of time : ) The best thing to do is to ask professors whose classes you enjoyed about their research once you get started at settled into university.

I am in my final year of an honours pure mathematics undergraduate and I can confidently say that all my fellow students who are thinking seriously about graduate school have participated in some form of summer research, or research during the year through classes offered by the department that...

It looks good : ) maybe someone else will spot something but there is no obvious flaws I have noted. I'll go through it in detail in a bit

This is more or less the same idea as the proof I did. the idea definitely makes sense although there are lots of details I haven't been able to check with your proof

Hahaha this is a great analogy. But seriously it can be spot on. Sometimes in your 'favorite" subjects you will spend too much time on the limited sub-topics you find most attractive. I don't think that this alone will account for "tanking" the class. Maybe the class you preferred just...

I am not sure what you mean by gaps here: U_1=\mathbb{R} by definition (I have used the fact that you can take the collection of open sets to be nested), and if you look at the definition of W_i it isn't too hard to see \mathbb{R}-A=\cup_{i\in \mathbb{Z}_+}W_i. Can you be more specific?

I am pretty sure it is sound but I will double check the proof when I am not at work, its been a few days so I might have missed something

That looks good to me verty and haven't found any problems in the proof up to this point. Good work :)