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Oct
10
comment Isn't it cheating to consider an $ \mathbb{R}^3 $ vector as a “pure quaternion”?
@Mariano: what is MU?
Oct
1
comment Square roots of $-1$ in quaternion ring
Ouch, I found my mistake. I must remember that the binomial theorem only works in commutative rings. Thank you!
Oct
1
asked Square roots of $-1$ in quaternion ring
Sep
28
comment What makes simple groups so special?
Related: math.stackexchange.com/questions/25315/…
Sep
28
answered Why is any compact metric space the union of a countable set a subset which is a perfect space under the induced topology?
Sep
27
comment Why is any compact metric space the union of a countable set a subset which is a perfect space under the induced topology?
See en.wikipedia.org/wiki/Cantor%E2%80%93Bendixson_theorem
Sep
19
comment Is there a more efficient method of trig mastery than rote memorization?
@GiuseppeNegro: I think you should expand your comment into an full-fledged answer, expanding on your claim. I would certainly vote you up ;)
Sep
16
comment free $R$-algebras: when does $R\langle X\rangle\cong\!R\langle Y\rangle$ $\Rightarrow$ $|X|\!=\!|Y|$ hold?
+1 for a well-posed, motivated question.
Sep
11
comment Motivation for Eisenstein Criterion
@Qiaochu: In Gallian's book, he states the "Mod p irreducibility test" with an additional hypothesis, which is that the reduced polynomial mod $p$ must have the same degree as the original polynomial in $\mathbb{Z}$ for the implication "irreducible over $\mathbb{Z}_p\Rightarrow$ irreducible over $\mathbb{Q}$" to work (and it seems an important point to the proof). Why don't you require it? Can it be proven without that hypothesis?
Sep
11
comment Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$
Related: math.stackexchange.com/questions/4764/…
Sep
8
comment Methods to see if a polynomial is irreducible
...if $a$ is a unit.
Sep
7
comment What's the index of a subfield?
As a comment, I'd like to point out that using the same notation for different notions (degree of a field extension and index of a subgroup) is useful, the Galois correspondence theorem being its prime evidence.
Sep
7
answered Finding $\lim\limits_{x \to \infty} \sqrt{9x^2+x} - 3x$
Sep
4
awarded  Nice Question
Sep
4
revised What's bad about left $\mathbb{H}$-modules?
added 233 characters in body
Sep
3
comment Quotient objects, their universal property and the isomorphism theorems
Thank you for your answer. What about considering a category with zero morphisms and normal monomorphisms? This would cover at least the category of groups. At any rate I'm still not fully satisfied, as per your comment on your last paragraph... maybe category theory is not the right language to express what is at hand?
Sep
2
revised What's bad about left $\mathbb{H}$-modules?
added 307 characters in body
Sep
2
comment What's bad about left $\mathbb{H}$-modules?
@Soarer: In general, you can define the tensor product of two $R$-modules over a commutative ring, and get a new $R$-module. If your ring is not commutative, then the tensor product does not naturally have the structure of an $R$-module, and thus remains only as an abelian group. If you accept bimodules, you can get a module structure without the ring necessarily being commutative, see en.wikipedia.org/wiki/Tensor_product_of_modules "Additional structure".
Sep
2
revised What's bad about left $\mathbb{H}$-modules?
added 74 characters in body
Sep
2
comment What's bad about left $\mathbb{H}$-modules?
@Soarer: sure, sorry, sometimes I first post a draft of the answer and then repeatedly edit it to add more content.