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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.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)##.
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, butA 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.
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.
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: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.