I What's the difference between graph, locus & manifold?

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fresh_42

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No. Closed alone makes no sense, i.e. needs a specification by an adverb or by context. You used closed as closed under arithmetic operations, but this is trivially given by definition of a field. In Galois theory we consider algebraically closed fields, which are fields in which any polynomial ##p(x)=a_n+a_{n-1}x+\ldots + a_1x^{n-1}+x^n## can be written as ##p(x)=(x-\alpha_1)\cdot \ldots \cdot (x-\alpha_n)##.
OK, I think I understand this now. If I have an established field F and I add a new number that is a root G(1/n) as a field extension to F, because I can do any of the arithmetic operations, adding the new number G is equivalent to adding any number that would result from using that number G(1/n) as the argument for a polynomial having coefficients that are in F. This is so because the only extra numbers that are added are those that have terms that contain the original numbers in F and any power of G(1/n), or G(k/n), with k being in the range [0,n) since the power of a root repeats as per a DeMoivre root around a circle.
 
A locus can be more than 1 dimension. Perhaps it is that a surface manifold has a 2-dimensional locus of points for the surface and 1-dimensional locus for the edge of the surface? This sounds like the idea of the surface integral used in Maxwell's Equations.
 

fresh_42

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A locus can be more than 1 dimension. Perhaps it is that a surface manifold has a 2-dimensional locus of points for the surface and 1-dimensional locus for the edge of the surface? This sounds like the idea of the surface integral used in Maxwell's Equations.
Sure. That was an illustration of the basic usage. I could as well have given you the correct definitions, but
  1. Wikipedia already has them.
  2. You don't read them anyway:
    I provided definitions of rings and fields in the other thread. If your research had been more careful, you would have seen that these terms didn't mean the same thing.
OK, I think I understand this now. If I have an established field F and I add a new number that is a root G(1/n) as a field extension to F, because I can do any of the arithmetic operations, adding the new number G is equivalent to adding any number that would result from using that number G(1/n) as the argument for a polynomial having coefficients that are in F. This is so because the only extra numbers that are added are those that have terms that contain the original numbers in F and any power of G(1/n), or G(k/n), with k being in the range [0,n) since the power of a root repeats as per a DeMoivre root around a circle.
This is also half baked and needed precision. The notation ##G^{1/n}## should be avoided, it is misleading. Maybe you should first examine how we constructed the following tower:
$$\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{Q}[\sqrt[5]{7},\sqrt{2}] \subseteq \mathbb{R} \subseteq \mathbb{C} \subseteq \mathbb{H}$$
and learn at each step what it is called: from half group to division ring.
 

mathwonk

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irresistible to try to answer such a vague question!

a locus is any set of points.

a graph (of a function) is a subset of a product space AxB which has at most one pair for each point of A.

the graph of a smooth function (defined on euclidean space and with values in a euclidean space) is an example of a (smooth) manifold.

In fact the previous example of a manifold is general in the sense that a (smooth sub-) manifold (of euclidean space) is everywhere locally just the graph of a smooth function.


this has no overlap whatsoever with stewart's galois theory.
 

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