Recent content by T-O7

Hello, I have two questions to ask regarding uniform convergence for sequences of functions. So I know that if a sequence of continuous functions f_n : [a,b] -> R converge uniformly to function f, then f is continuous. Is this true if "continous" is replaced with "piecewise continuous"...

Ah OK, I see. Thanks a lot! :smile:

Hey, There's one thought experiment that I can't seem to wrap my head around. If we had an extremely long rod (say > 1 lightsecond) and rotate it about one end rapidly enough (say 1 rad/second), would the far end of the rod be moving faster than the speed of light (using v=omega*r)...

Nice...that's wonderful. Thanks snoble! Exploit the isomorphism property, i get it. :smile:

Sorry...i don't know what you mean. I can come up with counter examples if it said "isomorphic as field extensions", because then i can just use the result that F(a) and F(b) are F-isomorphic iff a and b have the same irreducible poly. But I'm stuck in this case, with general field isomorphisms...

Hey all, I need to prove (or disprove) the following statement: F1 and F2 are two finite field extensions of a field K. Assume [F1:K]=[F2:K]. Then F1 and F2 are isomorphic as fields. Some help would be much appreciated. I know the statement is false if i replace "isomorphic as fields"...

Right...note to self, always remember about induction. Thank you. (although my solution using induction looks super messy)

Hey, Does anyone know how to show that these fields are equal: Q(\sqrt{p_1},\sqrt{p_2},...,\sqrt{p_k})=Q(\sqrt{p_1}+\sqrt{p_2}+...+\sqrt{p_k}), where p_1,...,p_k are distinct primes in Z. One inclusion is clear to me, but I'm having problems showing they're equal. Thanks!

Yes, great, thanks for the tip. After a little tedious work, it turns out that it is irreducible over F3, so my original polynomial was irreducible over Q. Yay, thanks a lot :biggrin:

Hmm..okay. I was hoping I wouldn't have to resort to brute force hehe OK, so I have shown that the polynomial x^6+x^3+2 has no linear or quadratic factors over F3, but how did you show it can't factor into cubic terms? I don't really know the irreducible cubic polynomials over F3, and it...

Hi, I need to figure out whether or not the polynomial x^6-2x^3-1 is irreducible (over Q). I don't think Eisenstein works in this case, and performing modulo 2 on this i get x^6-1 which is reducible over F2. Any ideas? Incidently, if i let y=x^3, then i get y^2-2y -1 which is...

Great, thanks a lot guys! Following Zurtex's method, i calculated: x^6-9x^4-8x^3+27x^2-72x-11=0 for x=\sqrt{3}+\sqrt[3]{4}. (There was a little error with zurtex's calculation i think) I managed to prove that this was irreducible tediously...but (this might be a dumb question) how did you...

Does anyone know how to find the degree over Q of this number: \sqrt{3} + \sqrt[3]{4} In fact I'm having trouble finding any generic polynomial that this satisfies! Please help :biggrin:

By rank i mean the maximum number of linearly independent rows/columns of the matrix. I guess you have to be careful about terminology when you're dealing with modules instead of vector spaces, I just find it a little confusing...I understand your point though, and i'll try to follow your line...

So if they are rank preserving, then the diagonal matrix A'' will have the same rank as A over Z, which is m (since the v's are linearly independent over Z), and since A'' clearly has rank m over R also, applying the inverses of the same integer elementary operations will give me back A, and I...