I found this problem in Do Carmos "differential geometry of curves and surfaces".
it asks to show that the length of a parallel curve B to A given by:
where r is a positive constant, and n is the normal vector, and A is a closed convex plane curve, positively oriented.
Very true! However, what does 2^(1/4) mean exactly in this case? i dont think it can be a real number since i dont believe there is an extension from F_5 to R...
And since there is no element x in F_5 such that x^4=2...
perhaps i am confused?
Thanks for you reply!
I was thinking about this,
finding the splitting field of x^4-2 in Q[x] over Q is standard enough... but would much be different is i wanted the splitting field over F_5? (field with 5 elements)
would it just be F_5(2^(1/4), i) analogously to the Q case? or do any of the arguments break down...
How would we find number of similarity classes for a nxn matrix over the field Fp (cyclic of order p) for n=2,3,4?
A and B are similar iff they have the same monic invariant factors
The Attempt at a Solution
I would say 4 classes for n=2, since...
ah so i see why it needs to be torsion free. (M lies in R, so if there is r such that rm=0, R cant be an integral domain...)
but what about the rank? Why must it be 1? I am also surprised that it is not free..
Hi all, I came across this problem in a book and I can`t seem to crack it.
It says that if we have an integral domain R and M is any non-principal ideal of R,
M is torsion-free of rank 1 and is NOT a free R-module.
Why is this true?
Hi all, i found this problem in a topology book, but it seems to be of an analysis flavour. I'm stumped.
Show that the collection of all points in R^2 such that at least one of the coordinated is rational is connected.
My gut says that it should be path-connected too (thus connected), but...
I was reading about the Nested sphere theorem and a thought occurred. if you have a sequence of decreasing closed sets whose diameter goes to zero in the limit,
we can show that the intersection of all these sets is a single point.
my idea was to show this using nested sphere theorem if we...
Origin of the term "seperable"
I was just curious as to why out of all properties of metric spaces (ie compactness, closure, etc), i dont know how the term seperable makes sense intuitively.... is there an origin to this term?