Search results for query: *

  1. M

    What am I doing wrong?

    Actually, I can't really see a problem. I really just that rusty? What's the answer supposed to be?
  2. M

    Algebraic expressions - simplifying

    It shouldn't be too terrible. The obvious thing to note is that (b - a) = -(a - b), for example.
  3. M

    Cyclic Subgroup of GL(2,q)

    It seems like you're going about the problem correctly except that obviously the splitting field won't be as described if the quadratic is reducible. Other than that, unless I'm forgetting something (which would hardly be surprising), what you're saying is completely true.
  4. M

    Matrix Multiplication

    In general. Well, for nonnegative integer exponents anyway.
  5. M

    Partial derivatives using definition

    The definition of the second partials is just the partial derivatives of the first partials. Why couldn't you just use the same method as before?
  6. M

    Fourier series via complex analysis

    Homework Statement Show that f is 2-pi periodic and analytic on the strip \vert Im(z) \vert < \eta, iff it has a Fourier expansion f(z) = \sum_{n = -\infty}^{\infty} a_{n}z^{n}, and that a_n = \frac{1}{2 \pi i} \int_{0}^{2\pi} e^{-inx}f(x) dx. Also, there's something about the lim sup of...
  7. M

    New Here. Question from power series

    Well, it looks like you're trying to find the power series of ln(5 - x) by differentiating the series for 1/(5 - x) term by term. But ln(5 - x) is the integral of 1/(5 - x) (give or take a sign).
  8. M

    More Abstract Algebra

    Oh, I meant for the intersection to be trivial. I'll think about what you said though.
  9. M

    Double integral with cos(x^n) term

    I think changing the order of integration is the way to go. You'll get an x^3 term in the integral with respect to x. Then it's easy.
  10. M

    More Abstract Algebra

    Homework Statement Show that G is isomorphic to the Galois group of an irreducible polynomial of degree d iff is has a subgroup H of index d such that \bigcap_{\sigma \in G} \sigma H \sigma^{-1} = {1} . Homework Equations The Attempt at a Solution I know that if G acts transitively as a...
  11. M

    Example in Abstract Algebra

    Homework Statement I'm trying to come up with an example of a quartic polynomial over a field F which has a root in F, but whose splitting field isn't the same as its resolvent cubic. Homework Equations The Attempt at a Solution Well, I know the splitting field of the cubic...
  12. M

    Triple integral over a sphere in rectangular coordinates

    I think you need to rethink your bounds on that one...
  13. M

    Trisectible angles | divisibility

    Without additional assumptions on m and n, the implications aren't true...
  14. M

    Complex analysis again

    Bump before bed
  15. M

    Trisectible angles | divisibility

    Isn't your first question essentially, "Can you construct an integer multiple of a constructable angle?" Well...can you?
  16. M

    Complex analysis again

    Homework Statement Let p(z) be a polynomial of degree n \geq 1. Show that \left\{z \in \mathbb{C} : \left|p(z)\right| > 1 \right\}[/tex] is connected with connectivity at most n+1. Homework Equations A region (connected, open set) considered as a set in the complex plane has finite...
  17. M

    Volume vs. Area of a Surface of Revolution

    Off the top of my head (that is, take this with a huge grain of salt), I think the approximating surface doesn't really matter in the answer, but the cone/cylinder might give the simplest (or maybe easiest to visualize) way to get to the answer.
  18. M

    Algebra questions, (emergency)

    1. First, the integers aren't even a group under multiplication. So you should be using addition as your operation. Under addition the integers are an (infinite) cyclic group. What does that tell you? 2. I'm not sure what you mean by mapping to an infinite number of items. But for the first...
  19. M

    Hints for finding a Galois closure

    I think I understand... The splitting field is the minimal field that contains all the roots of the minimal polynomial, and anything that's Galois over the rationals contains all the conjugates of \alpha (i.e., the roots of the minimal polynomial). So it contains the splitting field. Thanks...
  20. M

    Hints for finding a Galois closure

    I believe the splitting field is just the original extension adjoin i. So that's handy. But it seems like that really shouldn't be the Galois closure. Why would any Galois extension of the rationals that contains \mathbb{Q}(\alpha) have to contain i? It seems like it would only have to be a...
  21. M

    Hints for finding a Galois closure

    Homework Statement Find the Galois closure of the field \mathbb{Q}(\alpha) over \mathbb{Q}, where \alpha = \sqrt{1 + \sqrt{2}}. Homework Equations Um...the Galois closure of E over F, where E is a finite separable extension is a Galois extension of F containing E which is minimal...
  22. M

    Weierstrass M-Test and Absolutely Uniformly Convergence

    Well, I believe that to use the Weierstrass M-test you have to find a bound that works for all x on whatever set you're testing convergence in, not just a given x in the set. That is, your bound a is allowed to be a function of n, but it cannot be a function of x.
  23. M

    Weierstrass M-Test and Absolutely Uniformly Convergence

    Yeah, I'm pretty sure the Weierstrass M-test tests for uniform convergence by essentially testing for absolute uniform convergence...
  24. M

    Set Theory Proof: Proving lx-yl ≤2r for All x,y εA

    Here's a tip that will get you through a good chunk of analysis problems: add 0 inside the absolute value sign in a "creative" way.
  25. M

    Galois group of a polynomial

    Homework Statement Okay, I'm trying to explicitly determine the Galois group of x^p - 2, for p a prime. Homework Equations The Attempt at a Solution Okay, so what I've come up with is that I'm going to have extensions \textbf{Q} \subset \textbf{Q}(\zeta) \subset...
  26. M

    Set Theory Proof: Proving lx-yl ≤2r for All x,y εA

    No, that's not true. Just because |x - xo| is less than or equal to r doesn't mean that |x| + |xo| is too. Really, think more intuitively about what |x - xo| being less than r says.
  27. M

    Proving a^b Convergence from a_n and b_n Limits

    Well, if you don't mind a cheat-ish answer, you could just look at the logarithm, then use the facts that logarithmic and exponential functions are continuous and that products of convergent sequences converge to the product of the limits
  28. M

    The Center of a Ring and Subrings

    In general, a subring of R is a subset of R which is a ring with structure comparable to R. So you don't actually have to show all the axioms because multiplication being associative and distributive is inherited just by being a subset of R. Similarly, some of the additive group structure is...
  29. M

    Proving S∩T is a Subring of R

    What are the requirements for a subset of R to be a subring?
  30. M

    Being dense about an Algebra problem

    Yeah, I hate to keep doing this, but I still haven't figured it out. So I guess I'll try one more time. Sorry to be irritating.
  31. M

    Group homomorphism

    You're completely right. It doesn't imply g^8 = e. You're going to have to do a bit more work than just finding an element of the preimage of g'. Unfortunately, I'm having trouble coming up with the exact proof off the top of my head, so I can't give you much of a kick in the right direction...
  32. M

    Generation of isomorphic fields by separate algebraic elements

    Ah, of course. I must have been thinking of something else. I've got to stop posting when I haven't slept.
  33. M

    Being dense about an Algebra problem

    I guess I'll give this a quick bump before I go to bed. Maybe I was wrong about missing something simple and I actually do have to use the Galois condition.
  34. M

    Generation of isomorphic fields by separate algebraic elements

    Er...aren't those fields are isomorphic? Take a map that's identity on the rationals, and root 2 mapped to root 3, right? I mean, strictly speaking I don't think that the degree being the same is enough to guarantee an isomorphism (if I had to guess, I'd say the automorphism groups of the...
  35. M

    Please help w/ entire functions

    Wow, I'm really slow today. Okay, g(z) is not entire because, as you said, it's not analytic at 0. Why does your argument fail? Because the integral \displaystyle\oint_{c} g(w) dw is not defined if c passes through 0, so it's not 0 for every closed curve in the complex plane. However, since...
  36. M

    Please help w/ entire functions

    I'm pretty sure that what you have is \displaystyle\oint_{c} g(w) dw = \displaystyle\oint_{c} \frac{f(w)}{w} dw = 2 \pi i * n(c, 0)*f(0) So for that to be zero for all closed curves, f(0) = 0. Dunno if that helps any. And, as always, I could be wrong.
  37. M

    Test Question: Vector Proof Help

    Well, we can't really tell you if it said that you can't use unit vectors. Regardless, it's clearly true for any unit vector because, using your notation, the equation says ||v||^2 = v*v = ||v||. Edit: Yeah, I'm really out of it today.
  38. M

    Fall 2008 Grad Applications

    Grad. Don't ask me how I managed it. I personally vote for black magic.
  39. M

    Being dense about an Algebra problem

    Homework Statement I need to show that, given F \subset E \subset K \subset L (K/F is Galois but I don't know how important that is for the part of the problem I'm having trouble with) and a homomorphism \phi:E \rightarrow L that's the identity on F, that \phi(E) \subset K. Edit: Yeah, if it...
  40. M

    Identity map and injectivity

    By the way, my example map is completely wrong. That's what happens when you answer algebra questions while doing complex analysis I suppose.
  41. M

    Probably obvious complex analysis question

    Well, we haven't officially discussed path homotopy and its implications for contour integration yet (much to my chagrin), but I think I'm getting the idea. I think I could look at log(\frac{z+1}{z-1}), then maybe use that to define a square root in the region I want. Anyway thanks.
  42. M

    Inductive Proof on Well Known Sum

    So prove that (n(n+1)/2)^2 is equal to the right hand side. Much easier to verify.
  43. M

    Probably obvious complex analysis question

    I don't quite understand what a branch point is apparently. I mean, as I understand it, yes, -1 and 1 are branch points, but I thought that any point where the multi-valued "function" actually started to be multi-valued was a branch point. Oh well, -1 and 1 is the answer you're looking for I...
  44. M

    Inductive Proof on Well Known Sum

    Well, the second part isn't correct at all (\sum^{n}_{i=1} i^2 = \frac{n(n+1)(2n+1)}{6}), but that wasn't really my point. Look at the left hand side of your proposed equality.
  45. M

    Limits of integration

    I think it's just because t^2 = x doesn't necessarily imply that t = sqrt(x).
  46. M

    Probably obvious complex analysis question

    Well, log(z) has its branch points for z less than or equal to zero, so we get branch points for log(z^2 - 1) whenever z^2 - 1 \leq 0. So I guess it's -1 \leq z \leq 1?
  47. M

    Inductive Proof on Well Known Sum

    What is \sum^{n}_{i=1} i?
  48. M

    Probably obvious complex analysis question

    You interpreted correctly. Anyway, I think the branch points all lie on the real axis less than or equal to 1, assuming we're using the principal branch of the log.
  49. M

    Probably obvious complex analysis question

    That's how I got the "antiderivative" I mentioned, but if that's right then I really have no idea how to choose the right branch of the logarithm. Do I have to see what z + \sqrt{z^2 - 1} does to my path?
  50. M

    Is this integration probelm right so far?

    Infinity is just \infty. Dunno about the other. Anyway, it looks correct to me.