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  1. 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...
  2. U

    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]

    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?

    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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    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...
  6. 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...