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I am a graduate student at UCLA.


9h
comment Confused with imaginary calculus
$e^{z \log e}$.
9h
comment Confused with imaginary calculus
@Kyson Your last remark is correct. The power series defines a particular selection of values for $e^z$ which happen to form an entire function. But, for example, $e^{1/2}$ has two values, given by the positive and negative square roots of $e$. The negative square root is obtained by taking $\log e = 1+2\pi i$.
10h
comment Confused with imaginary calculus
They're not equal. This is because complex exponentiation does not define a single-valued function. What you've done here is found an alternate value for $1^{2\pi n i}$.
Jan
24
revised How many squares in a finite group?
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Jan
23
comment A walkthrough of how to apply Eisenstein's criteria to show that a multivariate polynomial is irreductible.
... and similarly for more than $2$ variables.
Jan
23
comment A walkthrough of how to apply Eisenstein's criteria to show that a multivariate polynomial is irreductible.
There are canonical isomorphisms $k[x,y] \simeq (k[x])[y] \simeq (k[y])[x]$: a polynomial in $y$ with coefficients in $k[x]$ can be distributed out into a polynomial in $x$ and $y$ with coefficients in $k$, and similarly for a polynomial in $x$ with coefficients in $k[y]$.
Jan
23
comment A walkthrough of how to apply Eisenstein's criteria to show that a multivariate polynomial is irreductible.
Not a complete answer, just a couple of quick points (I didn't read your reference, so this may be in there): $z^2+xy \in k[x,y][z]$ is irreducible by Eisenstein with $x$, for example. And $x^2+y^2+z^2$ can be shown to be irreducible as long as $k$ has characteristic $\neq 2$; otherwise $x^2+y^2+z^2 = (x+y+z)^2$. If $k$ doesn't have characteristic $2$, then $y^2+z^2$ factors iff $k$ has a square root of $-1$, but in this case the factors are not associates, so as remarked in your question 2, Eisenstein still applies.
Jan
23
revised Why Do The Axioms of Euclidean Geometry Not Need To Include the Definition of Space?
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Jan
23
comment Describe the subgroup $K\leq S_4$ of order 8
The elements of order 3 do not form a subgroup of $S_4$.
Jan
23
comment Finding $\lim_{x \rightarrow \frac{1}{4} \pi } \frac{\tan x-\cot x}{x-\frac{1}{4} \pi }$.
@NouvelRaka no problem!
Jan
23
comment Why Do The Axioms of Euclidean Geometry Not Need To Include the Definition of Space?
That misses my point: you don't need to describe an entire entity just to be able to describe the objects within the entity. Euclid's axioms have the advantage that they are both simple and powerful, hence to this day we still teach them to our children. More advanced geometry can be saved for students who are interested in studying that in higher education. And yes, everything you suggest has been done.
Jan
23
comment Finding $\lim_{x \rightarrow \frac{1}{4} \pi } \frac{\tan x-\cot x}{x-\frac{1}{4} \pi }$.
@NouvelRaka See my edit.
Jan
23
revised Finding $\lim_{x \rightarrow \frac{1}{4} \pi } \frac{\tan x-\cot x}{x-\frac{1}{4} \pi }$.
added 273 characters in body
Jan
23
answered Why Do The Axioms of Euclidean Geometry Not Need To Include the Definition of Space?
Jan
23
answered Finding $\lim_{x \rightarrow \frac{1}{4} \pi } \frac{\tan x-\cot x}{x-\frac{1}{4} \pi }$.
Jan
19
comment Hate calculus, but want to learn differential geometry?
Differential Geometry is literally calculus on manifolds. It's more theoretical in nature, so you might like it better, but I can't help feeling that not having any background in calculus might be a bit of a hindrance.
Jan
10
comment Show that $i^m + i^{m+1} + i^{m+2} + i^{m+3} = 0$ for all $m ∈ \mathbb N$
"Could someone please help me improve/shorten this answer?" Sure. Factor out $i^m$.
Jan
6
comment How the sequence ${1, 4, 9, …, n^2}$ can be written as sum of few arithmetic sequences?
Sounds like you're looking for $n^2 = \displaystyle\sum_{i=1}^n 2i-1$?
Jan
6
comment Convert any number to positive. How?
Hello. ${} {} {}$
Jan
3
comment generator of number field inside a given number ring
@Bernard ah, yes, I missed that.