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...
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...
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...
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...
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...
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...
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.
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...
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...
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...
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...
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.
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...
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...