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  1. U

    A (probably) simple combinatorial problem

    Thanks for your reply, Dick. I don't see how you can cover a circle of radius 4 by 3 circles radius 3 each, but 4 such circles is definitely enough. For instance, putting the center of the coordinate system at the center of the circle, so that it has equation x^2+y^2=4^2, the four small...
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    Isomorphism with z sub prime#

    Let f be an automorphism of Z_p, and x a generator for Z_p, so that <x>=Z_p. Explain why f is determined by what it does to x, i.e. why knowing f(x) suffices to to know where f sends any other element of the group. Now think about whether (x is a generator) => (f(x) is a generator) is true...
  3. U

    A (probably) simple combinatorial problem

    Homework Statement In a circle city of radius 4 we have 18 cell phone power stations. Each station covers the area at distance within 6 from itself. Show that there are at least two stations that can transmit to at least five other stations. Homework Equations The Attempt at a...
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    Kernel of GL(n,F) acting on F^n

    Homework Statement Suppose GL(n,F) acts on F^n in the usual way. Consider the induced action on the set of all k-dimensional subspaces of F^n. What's the kernel of this action? Is it faithful The Attempt at a Solution Well, I anticipate that the kernel of this action consists of scalar...
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    GF(q); x^p-x+a has no root => irreducible [PROOF]

    So, we know that, assuming that b is a root in some extension of F, then f(x)=\prod_{j=0}^{p-1}(x-(b+j)). Now suppose that f is reducible, that is f=gh for some polynomials g and h whose coefficients are in GF(q). There must exists a proper subset I of {0,...,p-1} such that...
  6. U

    Partial derivatives and chain rule

    I apologize. The solution is correct though.
  7. U

    GF(q); x^p-x+a has no root => irreducible [PROOF]

    OK. It looks that there is some notation misunderstanding. I'm not that well familiar with finite fields so I might have used the type of notation not usually used. By GF(q) I mean a finite field of order q (with q elements) where q is a power of prime p, and that p is a characteristic of...
  8. U

    Partial derivatives and chain rule

    d/(dv)[du/ds]= d/(dv)[du(s,t(s,v))/ds]= d/(dv)[du/ds + du/dt * dt/ds]= d/(dv)[du/ds+du(s,t(s,v))/dt * dt(s,v)/ds]= d^2u/dsdt * dt/dv + d^2u/dt^2 * dt/dv * dt/ds + du/dt* d^t/dsdv Write it down in usual partial derivative notation and you'll see where it all comes from.
  9. U

    GF(q); x^p-x+a has no root => irreducible [PROOF]

    Homework Statement Let q=p^e, where p is a prime and e is a positive integer. Let a be in GF(q). Show that f(x)=x^p-x+a is irreducible over GF(q) if and only if f(x) has no root in GF(q) Homework Equations The Attempt at a Solution One of the directions seems obvious. Namely, if...
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    Char(F)=p; poly NOT irreduc. => must have a root?

    Why are you referring to a root of x^p-a as THE root? How do you know there is only one root? What if they behave like roots in the complex field?
  11. U

    Char(F)=p; poly NOT irreduc. => must have a root?

    How about these arguments. All the roots of x^p-a are p-th roots of a. Suppose that x^p-2 is not irreducible and factorize in into irreducibles. All of the will be of degree >=2. Consider any of them. Look at the constant term of that factor. It MUST (BUT WHY?!) look like the p-th root of a to...
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    Char(F)=p; poly NOT irreduc. => must have a root?

    First of all, it is not true that x^p-2 can be represented as a product of only two irreducible polynomials.
  13. U

    Char(F)=p; poly NOT irreduc. => must have a root?

    If it has a root everything is clear. What is not clear is why the fact that x^p-a is NOT irreducible implies that it has a root.
  14. U

    Char(F)=p; poly NOT irreduc. => must have a root?

    Because it is indeed correct? In a field F of characteristic p for any two x and y in F we have (x+y)^p=x^p+y^p. This is to see using the mentioned binomial expansion and convincing yourself that p divides every binomial coefficient except the frist and the last. Thus, each term in the expansion...
  15. U

    Char(F)=p; poly NOT irreduc. => must have a root?

    Homework Statement Let F have prime characteristic p and let a be in F. Show that the polynomial f(x)=x^p-a either splits or is irreducible in F[x]. I was given a hit: "what can you say about all of the roots of f in a splitting field?" Homework Equations The Attempt at a Solution...
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    Prove the sum of squares of two odd integers can't be a perfect square

    Note that the sum of the square of two odd integers (2m+1) and (2n+1) is X=4(n^2+m^2+n+m)+2. Clearly, X is congruent to 2 modulo 4. But k^2 can only give remainder 0 or 1 modulo 4.
  17. U

    Equal additive order of all elem in simple ring

    Homework Statement Let S be a simple ring. Show that all nonzero elements of S have equal additive order. Show that this order either is a prime number p or is infinite. The Attempt at a Solution All I could show is that the order of any element x in S must divide that of the unity...
  18. U

    R-UFD, F-fraction field, f monic in F[X], f(a)=0=> a in R

    Homework Statement Let R be a UFD and let F be the field of quotients/fractions for R. If f(a)=0, where f is in R[X] is monic and a is in F, show that a is in R. The Attempt at a Solution On one hand if a is F, we can write a=m/n for some m and n in R, where in fact m and n are comprime. Then...
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