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