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"...
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)...
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...
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"...
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...
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...
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.
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)...
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.
.....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...
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...
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...
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...
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...
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:
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...
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...
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...
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: