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Jan
22
comment Distributive Law and how it works
My usual response to questions of this form, i.e. “Why isn’t it such-and-such a way?” is to say that In mathematics, nothing is true. Except when there’s a proof. There are proofs for the proper distributive rule, in various contexts, and of course there’s all the experimental evidence.
Jan
21
comment Inseparable extensions
So you’re reduced to the case that $F$ is purely inseparable (“radicial”) over $K$, say of degree $p^n$. Now let $a\in F\setminus K$, so its $K$-irreducible polynomial is $X^{p^m}-b$ for $b\in K$. If $K(b)=F$, you’re done, and if not, then pull the same trick on $K\supset K(b)$, inductively. This should work.
Jan
21
comment Is there a better representation than p-adics for exact computer arithmetic?
Umm, “even if I choose $p=2000000001$” — if you’re using decimal notation, that’s obviously divisible by $3$.
Jan
21
comment ducci sequences using p-adic evaluation
But the map doesn’t send $\Bbb Z_p^n$ to itself.
Jan
21
comment Will someone kindly explain Kato's dual exponential map?
To get any help, I think you need to expand considerably, explaining all the notations. Otherwise, you depend on that fraction of the MSEers who are intimately familiar with the paper.
Jan
21
comment Inseparable extensions
What have you tried? Are you aware that there’s a maximal separable extension of $K$ in $F$?
Jan
19
comment basis of the p-adics $\mathbb{Q}_p$ as a $\mathbb{Q}$-vector space
When it comes to logic, I’m an outsider looking in, but I find this answer very, very pleasing.
Jan
19
comment $\epsilon - \delta$ proof of limit of $\arctan \left(\frac {x+z}{y}\right)$ as $(x, y, z) \to (1, 2, -3)$
I tried to cook up a slick proof using the fact that $\arctan$ is a contractive map (i.e. $\arctan(x+\varepsilon)-\arctan x<\varepsilon$ for positive $\varepsilon$), but my efforts rapidly became unslick.
Jan
19
answered What is the most used method for proving continuity for simple functions such as $f(x) = x^{1/3}$
Jan
18
comment Discrete valuation fields and representation as power series
Yes, the comment of @Mathmo123 may well be pointing out the source of your misunderstanding.
Jan
18
comment Discrete valuation fields and representation as power series
Then I have to ask: have you found a pair $f$ and $g$ such that the sum or product is different in the two rings? Or a sequence that’s convergent in one but not the other? The admitted fact that the two definitions look very different is not sufficient for concluding that the two are different. I find your question reasonable, even though I disagree with you, because I know of a parallel situation where two things look alike but are different, it’s the case of power series over $\Bbb Q_p$ and look at the $p$-adic topology induced from $\Bbb Z[[x]]$, and the coefficientwise topology.
Jan
18
comment Discrete valuation fields and representation as power series
I’m not sure I understand. Are you observing that the definitions seem very different, but the results are the same? Or are you fearful that the results are in fact different?
Jan
18
comment Is there a better representation than p-adics for exact computer arithmetic?
Sorry, can’t help you there: I’m a number theorist (of sorts), not a computer scientist, and for representation of rational numbers, using arbitrary-length integers is good enough for me, the problems associated with that representation notwithstanding.
Jan
18
comment Is there a better representation than p-adics for exact computer arithmetic?
As a $p$-adic guy, I find the question very interesting from a philosophical standpoint, but not so much from a practical. Mainly because the $p$-adic numbers I’m interested in are rarely rational, and it would be an unusual problem that required more than a dozen $p$-adic places of accuracy.
Jan
18
comment Find the angle between planes given three lines
Yes, it’s relatively quick, certainly easily programmed, and has the advantage (over anything using the spherical Law of Sines) that since your angles will always be between $0$ and $180^\circ$, the $\arccos$ function is one-to-one and will give a reliable answer.
Jan
18
answered A function in the integers module $p$ is polynomial.
Jan
18
revised Find the angle between planes given three lines
give proper credit to a commenter
Jan
18
comment Find the angle between planes given three lines
I hope you don’t mind, @achillehui, that I have expanded your comment to an answer.
Jan
18
answered Find the angle between planes given three lines
Jan
18
comment Group law on elliptic curves.
It’s the same reasoning that tells you that if $f$ is a cubic polynomial over $k$ with two roots in $k$, then the third root will also be in $k$.