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16h
answered For extension fields, does $[F(a,b):F(a)]=[F(b):F]$?
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revised Identifying the algebra
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revised Identifying the algebra
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comment Identifying the algebra
This follows from Bergman's diamond lemma applied to the lexicographic monomial order with $c>b>a$ and the relations I wrote.
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answered Identifying the algebra
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comment $C(X)$ is separable when $X$ is compact
@AdamHughes, your hint does not mean anything in this context, as the domain of the functions is not a subset of a field...
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comment $C(X)$ is separable when $X$ is compact
Your argument for part $a$ does not make sense unless you tell us what you mean by $\perp$.
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comment Is there a group-theoretic proof of the Riemann rearrangement theorem?
@ThomasAndrews, such permutations are called shuffles and the way they compose is (in the finite case, at least) very, very interesting and useful in many contexts :-)
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comment Number of ways distribute 12 identical action figures to 5 children
There is no way to distribute 12 things to 5 kids that will not cause problems.
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comment When are two morphisms of sheaves the same?
A sheaf of groups is an abelian group object in the category of sheaves of sets, Martin. As I am talking to humans, the "underlying" part is essentially just noise.
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comment When are two morphisms of sheaves the same?
Sheaves of abelian groups are sheaves of sets!
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comment When are two morphisms of sheaves the same?
If you add hypotheses, you'll get more lucky: flasque or soft or...
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comment If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic
@Bryan, that identity is essentially known as Moebius inversion and it is extremely useful!
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answered When are two morphisms of sheaves the same?
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comment When are two morphisms of sheaves the same?
You can well have $F(X)=G(X)=0$, and in that case your condition does not give much!
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awarded  Enlightened
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awarded  Nice Answer
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comment Given basis for a Lie algebra, what is one for its Universal Central Extension
In particular, there may be one or several central terms in the extension, and they are described by cohomology.
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comment Given basis for a Lie algebra, what is one for its Universal Central Extension
The central extensión is completely determined in terms of the cohomology of the álgebra. You could do worse than read the relevant section in Weibel's book on homological algebra.
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comment Given basis for a Lie algebra, what is one for its Universal Central Extension
Af for what is a basis of the universal extension, well... it depends on the Lie álgebra! I cannot imagine what sort of answer you expect here.