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Apr
11
awarded  Nice Question
Apr
11
awarded  Nice Answer
Apr
8
comment why can't quintics be solved by radicals and the relevance of permutations of roots of polynomials
perhaps you would like to read Galois' original text as well.... Joking aside, the modern treatment through Galois theory is an enormous simplification over any of the original treatments. Tackling, say, Abel's work directly is going to be extremely challenging, while there are several excellent books that go from nothing to the Galois correspondence in under 150 pages, complete with all the linear algebra and group theory prerequisites, and applications not only to the insolubility of the quintic, but (why not, it's so easy once you have Galois) other famous results.
Apr
6
revised Prove that $A^k = 0 $ iff $A^2 = 0$
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Apr
3
comment How many quadratic extension are there on a field?
Do you think that $\mathbb Q (\sqrt 2) $ is isomorphic to $\mathbb Q (\sqrt 3)$? Try to construct an isomorphism.
Apr
3
answered limit functors as adjoints
Mar
29
comment Why is cardinality of set of even numbers = set of whole numbers?
@nxs responding to your comment to me, you seem to confuse abstract with arbitrary. There is nothing arbitrary in the choice of axioms for the things we study, no matter how abstract or concrete they are.
Mar
29
comment Why is cardinality of set of even numbers = set of whole numbers?
@nxs your confusion stems from trying to think of cardinal equality in geometric terms. That simply can not be done since by its very definition cardinal equality is about abstract sets, devoid of any geometry. You got confused because you consider sets for which you have a concept of geometry for, and you forgot to neglect it since you are only concerned with cardinality.
Mar
25
revised What is the difference between topological and metric spaces?
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Mar
22
revised Is the Cartesian product of sets associative?
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Mar
22
comment Why study dimensions?
measures 'count' side in a different way. The rationals for instance are countable, so are small relative to the reals, i.e., they have measure 0. But the closure of the rationals are the reals, so in that respect they are large with respect to the reals, i.e., they are dense. Size matters, and size may change according to perspective.
Mar
22
answered Why study dimensions?
Mar
22
revised Empty intersection and empty union
deleted 1 character in body
Mar
18
revised do the uniformly continuous functions to the reals determine the uniformity?
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Mar
15
comment noncommutative ring with unity..
is any of the sets you describe closed under addition and multiplication? (hint: no!).
Mar
14
answered How do you show one way equivalences in mathematics?
Mar
13
comment The distance between disjoint closed sets may be zero
Welcome to MSE. This is not a drop-your-homework-on-us type site. What have you done to try and solve this problem? where did you get stuck?
Mar
13
comment How to call a mathematical space “$(\mathcal S, f)$” consisting of set $\mathcal S$ and function $f : \mathcal{S \times S} \rightarrow \mathbb R$?
Good luck!! If you can obtain interesting results about these 'spaces', then you can post some of your results and motivation, and seek ideas for a name. At this point, it's an unborn baby, one that was not even conceived yet. Don't get too attached to it.
Mar
13
comment Is it possible to prove uniqueness without using proof by contradiction?
@MathematicsStudent1122 certainly, though that is an overkill as it follows easily by elementary properties of the reals.
Mar
13
comment Is it possible to prove uniqueness without using proof by contradiction?
It would be if you provided a proof of the injectivity of the function that does not use contradiction. Right now, all you stated is a tautology. You did not prove anything. Moreover, you say "we use the injectivity property" but then you go ahead and seem to deduce the injectivity of the function". It is not clear what you are trying to do in this answer.