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Jun
25
comment Simplicity is invariant under extension of scalars
@rschwieb Suppose $\sum_k a_k\otimes\xi_k\in Z(A\otimes_k K)$ and $\xi_1,\dotsc,\xi_n$ are linearly independent over $k$, and it commutes with any $b\otimes\eta$, then by simple calculation it could be shown that $a_k$ commutes with $b$ for any $b\in A$, thus $a_k\in k$ and thus $\sum_k a_k\otimes\xi_k\in K$.
Jun
24
comment Simplicity is invariant under extension of scalars
@rschwieb Well, I need to check whether that $A$ is central $k$ implies that $A\otimes_k K$ is central over $K$. I proved this, but I'm not 100% sure. I didn't touch with extension of scalars before.
Jun
24
comment Simplicity is invariant under extension of scalars
@JeremyRickard You're right. I misunderstood the sentence.
Jun
24
asked Simplicity is invariant under extension of scalars
Jun
23
comment Understanding calculus formulas intuitively
@KCd Zorich's is good.
Jun
22
comment The Galois group of a composite of Galois extensions
Well, thanks. I've been away from Galois theory since then, and I still find that I don't digest Galois theory well. Many theorems are still nontrivial to me, so I need to refresh the materials. Infinite Galois theory and profinite groups are still left indigestions, which I want to compare with ring completions, also an application of inverse limit. Incidentally, is there any material on (elementary) geometric (especially, algebro-geometric) viewpoint towards Galois theory?
Jun
22
comment The Galois group of a composite of Galois extensions
Well, I'm looking for some experts to check this proof, so the acceptance might be postponed. Sorry!
Jun
19
comment The Galois group of a composite of Galois extensions
It's an ancient post and I feel glad that I eventually receive an answer. It would be better if you can furnish some good reference for infinite Galois theory. It seems that the corresponding chapter of Morandi's book is indigestive to me.
Jun
14
comment Function coincides with a function of bounded variation almost everywhere
Incidentally, could you write out your original proof which makes use of approximations to identity for reference or historic reasons?
Jun
10
comment On existence of a tangent line passing through a given point
But maybe all tangent lines corresponds to the Zariski closure of the corresponding points to tangent lines at nonsingular points. I guess that the closure is irreducible and then Bezout's theorem works. I don't know whether it is true.
Jun
10
comment On existence of a tangent line passing through a given point
@AsalBeagDubh How to deal with singularities? The dual curve is usually defined as the image of $(\partial f/\partial x:\partial f/\partial y:\partial f/\partial z)$, which isn't well-defined at, say, double points.
Jun
10
revised On existence of a tangent line passing through a given point
edited tags
Jun
10
asked On existence of a tangent line passing through a given point
Jun
1
comment If R(z) and S(z) be two rational functions, if $\forall |z|=1,R(z)=Q(z) $, how to prove $R(z)\equiv Q(z)$?
Note that if two polynomials agree on an infinite set, then they're identically equal.
May
30
comment Height and minimal number of generators of an ideal
See here for your last comment.
May
24
revised Are maps locally preserving collinearity homographies?
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May
20
comment Are maps locally preserving collinearity homographies?
By the way, it's unfair that such a good answer didn't accept desirable upvotes.
May
20
comment Are maps locally preserving collinearity homographies?
I think it even works for any domains (connected open regions), since $\mathbb R^2$ is locally path-connected and we can use consecutive discs to cover the path connecting two points. I haven't written up the rigorous proof.
May
18
comment Why is the projection map proper?
Okay, edited. I hope you can check it.
May
18
revised Why is the projection map proper?
added 1226 characters in body