Recent content by snoble

There are equivilant definitions of Galois extensions listed here http://mathworld.wolfram.com/GaloisExtensionField.html but I'm confused about the equivilence of 1 and 2. What am I doing wrong here? Take K to be the splitting field of X^4-2 over \mathbb{Q}. This is exactly property 1. But...

The number of ways n people can have birthdays is simply 365^n. To be honest I didn't really understand your argument for why you could use 2C6 but I would guess your problem is you assumed things extended when there is infact no justification for it. Also the number of ways for n people to...

Ok guys thanks for thinking about this one but I finally found a reference that proves it by a guy named Cassels. Turns out you need a bit more care in picking your prime... you need the poly to factor and the prime not to divide the discriminant. Then you can use a corollary to Hensel's...

close a_n\cdot (z-c_1)(z-c_2)(z-c_3)...(z-c_n)=a_nz^n+a_{n-1}z^{n-1}+...a_0z^0

I know of exactly what you are thinking about Hurkyl and I too have forgotten their names. However the key is that they are multivariate. I think they are important to a certain type of invariant theory. But integral polynomials over a single variable do factor. Of course the question is do...

Ok, good. That was the part that I found dubious as well. Here is how I justified that to myself (though I am no way convinced by this justification). The roots of the original poly being in a primitive expansion means that for each root there is a polynomial (well rational function)...

Does this argument sound reasonable? A poly will split completely in some field if that field contains all it's roots. Any algebraic extension of the rationals is a subset of a primitive extension. Just consider the minpoly of that primitive element (using the algebraic extension...

But couldn't that be a field who's order is a prime power instead of just a prime?

Can I suggest checking out the mathworld page http://mathworld.wolfram.com/RiemannZetaFunction.html. Since the zeta function has become so popular in popular mathematics mathworld seems to have gone out of its way to write a very thorough discussion including a discussion of analytic...

Certainly the geometric method is the most straight forward but I didn't suggest it because the question asked about using complex numbers. The geometry of the trig functions can be seen while completely ignoring complex numbers. When it comes to student exercises clearly the process is more...

Does anyone know if this is true and if so where they know it from? Given a polynomial over the integers there exists a finite field K of prime order p, such that p does not divide the first or last coefficient, and the polynomial splits over K. I realize this could be considered an...

I haven't seen this type of problem before but here's what I'm thinking. Take cos\theta =p,\, sin\theta=q. So expand out (p+iq)^5=1. So the imaginary part is 0 and the real part is 1. So collect the multiples of i and substitute [itex]q=\sqrt{1-p^2}[/tex]. You'll end up with something you...

Hmm, connectivity does seem to be an issue. A friend pointed out to me that this is based on the analytic identity theory which basically says if two functions are analytic on a domain and they agree on a set with an accumulation point then they are equal on the domain. Connectivity seems to...

and perhaps a reference. Given a complete normed space S (metric space may be sufficient, I'm not sure) with a compact subset C and a function f that is analytic on C then if f(x)=0 for infinitely many x\in C then f is identically 0 on C. I'm sure I learned this in an undergraduate...

hmm... I think problem 2 is just fine without any added assumption. Well you have to assume friendship is mutual. If a is a friend of b then b is a friend of a. And no one is their own friend. But these are reasonable assumptions. So my two big hints are: 1)there is nothing special about...