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    Uniform convergence

    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"...
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    A really really long rod

    Ah OK, I see. Thanks a lot! :smile:
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    A really really long rod

    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)...
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    Field extensions

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

    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...
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    Field extensions

    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"...
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    Are these two fields equal

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

    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!
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    Is this irreducible?

    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:
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    Is this irreducible?

    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...
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    Is this irreducible?

    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...
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    Degree of this number

    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...
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    Degree of this number

    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:
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    Linear independence over Z

    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...
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    Linear independence over Z

    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...
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    Linear independence over Z

    Ok, i think i see what you're saying....in getting the diagonal matrix A'' from A, i had tacitly assumed that the elementary integer matrices are rank-preserving over Z.....and i really can't find a way to convince myself whether or not that's true. I'm going to try your outline now, i think...
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    Linear independence over Z

    Yeah, i should have explained earlier, this theorem was discussed in class, and I suppose this question was asked in hopes of understanding that the theorem was applicable.
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    Linear independence over Z

    But there's a theorem at my disposal: if we assume the v's are independent over Z, and A is the matrix whose columns are the v's, then through a series of elementary integer matrix multiplications we get a diagonal matrix, of the form A' = \left(\begin{array}{cc}D&0\\0&0\end{array}\right)...
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    Linear independence over Z

    Hmm...now that i think about it, i don't think i can give an argument using determinants anyway (i.e showing that the determinant is non-zer0), since i can't assume that m=n.
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    Linear independence over Z

    .....hmph. So the fact that a n×n matrix is invertible iff it has rank n is false over Z? I think i might've also made the mistake of assuming m=n in my argument above...
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    Linear independence over Z

    Ah, I hadn't thought of that! So following what matt suggested, if A is the matrix whose columns are the v's, we get a diagonal matrix through a series of elementary matrix multiplications, which don't affect the rank of A, and so if the columns are independent (over Z), then A must have...
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    Linear independence over Z

    okay, i get a little confused even with the case m=1. For the case m=1 we consider the vectors v as just elements in Z, and the question is: if they're linearly independent over Z, are they linearly independent over R? My issue right now though is that they can't even be linearly independent...
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    Linear independence over Z

    Hey all, I need to show whether or not the following statement is true: For v_1,...,v_n\in Z^m, the set \{v_1,...,v_n\} is linearly independent over Z \Leftrightarrow it is linearly independent over R. The reverse direction is true of course, but i'm having some trouble showing whether or...
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    Stereographic projection

    So i'm trying to prove that the map f(x,y,z) = \frac{(x,y)}{1-z} from the unit sphere S^2 to R^2 is injective by the usual means: f(x_1,y_1,z_1)=f(x_2,y_2,z_2) \Rightarrow (x_1,y_1,z_1)=(x_2,y_2,z_2) But i can't seem to show it.... :frown: I end up with the result that...
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    Equation of evolute?

    Yeah, i made a boo boo calculating the arclength :redface: . And apparently the definition i was given for the evolute of a unit-speed curve can also be used for any other parametrization of the curve, so everything's a-okay. :smile:
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    Primes in ring of Gauss integers - help

    ah, so a^3=1, and a has order 3. We can apply the argument backwards, and that will prove a). I see, thanks Hurkyl! :smile: There's also a second part to this problem, which says (p) is prime ideal in R if and only if p=-1 (mod 3) Apparently the first part of this problem applies, but i'll...
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    Primes in ring of Gauss integers - help

    Hmm....sorry, i don't see what you mean.
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    Primes in ring of Gauss integers - help

    Primes in ring of Gauss integers - help!! I'm having a very difficult time solving this question, please help! So i'm dealing with the ring R=\field{Z}[\zeta] where \zeta=\frac{1}{2}(-1+\sqrt{-3}) is a cube root of 1. Then the question is: Show the polynomial x^2+x+1 has a root in F_p if...
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    Equation of evolute?

    Ok, so i need to calculate the equation of the evolute for the catenary \gamma(t)= (t,cosh(t)). I'm not really sure how to do this, the definition of evolute I have requires a unit-speed parametrization, but it looks a little difficult to find that also (the arclength if i'm correct is...
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    Is this a field?

    Arghh....yes, you were right. I had overlooked these four: x^2+2, x^2+3, x^2+x+2, x^2+4x+1 And you were right again, x^2+1 did divide x^4+1. Well, now I guess it's either: x^4+2 or x^4+3. One of them's gotta work right? :bugeye:
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