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838103
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location USA
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visits member for 2 years, 4 months
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PhD

Interests:

Ring and module theory

Clifford algebra/Geometric algebra

Mathematical physics

Applications of abstract algebra


12h
comment Show that there is no surjective ring homomorphism from $\mathbb Z_2[x]$ to $\mathbb Z_2 \times \mathbb Z_2\times \mathbb Z_2$
The long story short is that the second ring has three quotients isomorphic to the field of two elements, but the first ring only has two.
1d
comment What is a geometric structure?
@Buddha This is why I included the link to the "is category theory geometric?" question. Every category can be viewed this way as objects with morphisms between them that preserve the salient features of each object. At that level, yes, it can be considered geometric. I don't think one would go about defining geometric objects at that level of generality, though
1d
comment What is a geometric structure?
@Buddha So giving $\Bbb R$ the discrete topology, every function out of this space is continuous. Would you consider the characteristic function of the irrationals "smooth" geometrically? (You might consider it so, but this example kind of highlights the difference between "shape" and "continuity")
1d
comment What is a geometric structure?
You might be interested in this question: math.stackexchange.com/questions/896846/…
1d
comment What is a geometric structure?
The definition of a topology is based on subsets of the powerset because topology says something about the "internal organization" of the set. The Kleinian perspective on geometry is that you're observing a fixed set of transformations on a set, and seeing what is invariant. Since a good notion of geometric structure would have to invite the Kleinian perspective to the party, I'm not sure how one would unite the two under one definition.
1d
comment Proof that geometric product is associative
@ahala I don't follow what you're saying, but that's ok.
1d
revised Proof that geometric product is associative
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1d
revised Proof that geometric product is associative
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1d
answered Proof that geometric product is associative
1d
revised Midpoint of chord of contact
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1d
revised Midpoint of chord of contact
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1d
comment Does every free $R$-module have a maximal proper submodule?
@user2902293 While every free module has a maximal submodule, we unfortunately do not have enough control to find maximal submodules containing a proper submodule of our choice! I added an answer I remembered by Jack Schmidt that is related.
1d
revised Does every free $R$-module have a maximal proper submodule?
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2d
revised Midpoint of chord of contact
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2d
answered Does every free $R$-module have a maximal proper submodule?
2d
comment What is the motivation for quaternions?
@JyrkiLahtonen Hmm that's interesting: can you point me toward consructions of these finite dimensional division algebras over $\Bbb Q$? With all the finite field extensions of $\Bbb Q$ available, I guess I never sat down to think about the division ring extensions :)
2d
comment What is the motivation for quaternions?
Frankly, after the complexes, the idea that solving equations is the primary motivation for number systems breaks down. You can be forgiven for thinking that since apparently that seems to be the primary motivation given in schools...
2d
revised What is the motivation for quaternions?
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Aug
27
comment Sigma-Algebra: Is it an Algebra, Field, or Something Else?
@Mathemanic In the most loose sense, an algebra is a set with operations, so the usage in measure theory is not completely inappropriate.
Aug
27
revised Midpoint of chord of contact
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