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Apr
15
comment Power series in p-adic integers
Yes, you really do have to read up on all of this. I would draw the Newton Polygon of the log. This tells you the convergence-radius, and tells you that there’s a zero with valuation $1/(p-1)$. Closer contemplation of the NP shows you that the inverse of log can not be convergent as far as $\{z: v(z)=1/(p-1)$. Then you notice that conjugating by the homothecy $z\mapsto p^{1/(p-1)}z$ turns the log into a series with all $p$-integral coeffs, so that it now has an inverse valid on the open disc. This shows that the exponential is convergent exactly for $v(z)<1/(p-1)$. No factorials need apply!
Apr
15
comment Symbol to denote the angle between two points
I don't think your question is well posed, ’cause two points don’t determine an angle. If you mean the angle between the line determined by your two points and a horizontal line, then you might use $\arctan m$, where $m$ is the slope of the nonhorizontal line. You know how to denote that, in terms of the coordinates of your points.
Apr
12
comment A doubt about real analysis concerning countable sets
As @Tryss says, if, for you, “countable” implies “infinite”, then it’s not true that an injection from $A$ to $\Bbb N$ implies countability of $A$.
Apr
12
comment Roots of Unity for Precalculus
It might be useful to note that $x^4+x^2+1=(x^2+x+1)(x^2-x+1)$.
Apr
12
comment Finite extnsion of local field.
I don’t feel that you have answered my question. Does $K$ include the maximal unramified extension of $F$?
Apr
12
comment Finite extnsion of local field.
To me, it looks like a mistake in the paper. Probably not essential, since the kernel of that map consists of no more than the $p$-th roots of unity. You can check it out for the case $p=2$, where of course $-1$ goes to the identity in $\Bbb F_2$.
Apr
12
comment Finite extnsion of local field.
It’s unclear to me what you mean by “maximal tamely ramified extension of $F$”. Do you mean an extension that is totally ramified, and tamely so? Or do you mean an extension in which the only ramification is tame? The latter includes the maximal unramified extension of $F$, the former does not.
Apr
12
comment Is the algebraic closure of $\mathbb Q$ in the field $\mathbb Q_5$ a number field?
Hensel’s Lemma is deep and important, but not hard; if you want to understand $\Bbb Q_p$, you need to know it.
Apr
12
comment Why is $\frac{1}{3}=0.2\overline{31}$ in 5-adic expansion?
I like running the $p$-adic digits to the left: $...1313132;\,$. In this way, you can do hand computations exactly as you did in elementary school, using $p$-ary addition and multiplication tables as aid. Of course that applies to $+,-,\times$ only. Division has to be tabulated differently, but it’s easy enough to figure out.
Apr
10
comment Find the value of Taylor coefficient $a_5$ for the function $ \int_0^x (e^{-t^2}+\cos t) \, dt$
Yes, repeated differentiation of $\exp(f(x))$ gets dreadful very quickly unless $f$ is linear.
Apr
10
comment Find the value of Taylor coefficient $a_5$ for the function $ \int_0^x (e^{-t^2}+\cos t) \, dt$
In strong disagreement with @BGreen, I think that Rolf’s answer is the right one: you come to a series expansion of your function without doing any differentiation at all. Just substitute $-t^2$ for $t$ in the expansion of $\exp(t)$, and use what you know about the cosine series, and finally integrate term by term.
Apr
9
comment prove n divides $[\mathbb{F}[\alpha]:\mathbb{Q}]$
Don’t you have a chain $\Bbb Q\subset\Bbb Q(\alpha)\subset\Bbb F(\alpha)$? What is $[\Bbb Q(\alpha) : \Bbb Q]$?
Apr
8
comment Find the Laurent Expansion of $f(z)$
Definitely the quickest way to do it, and the best, in my opinion.
Apr
8
comment Irreducibility in $k((t))[y]$
Yes, to both comments. The upshot is that in most cases, $f(y)-t$ will be reducible.
Apr
8
comment What method was used here to expand $\ln(z)$?
It’s missing the even terms because it’s an odd function: replace $z$ by $-z$ and your fraction turns upside down, so its logarithm changes sign as well. PS: the Grammar Pedant says that the word in this case is not “jive” but “jibe”.
Apr
8
comment Irreducibility in $k((t))[y]$
I don’t believe that the characteristic is a significant part of this story.
Apr
8
comment Algebra: Field Theory Question
Yes, that is indeed true.
Apr
8
comment How to show that the ring of integers of $\mathbb Q(cos\frac{2\pi}{m})$ is $\mathbb Z(2cos\frac{2\pi}{m})$?
I’m sure you’ve noticed that $2\cos(2\pi/m)=\zeta_m+\zeta_m^{-1}$, where $\zeta_m$ is the primitive $m$-th root of unity $\exp(2i\pi/m)$. I think that if you know how to show that $\Bbb Z[\zeta_m]$ is the integers of $\Bbb Q(\zeta_m)$, then the proof of your proposition should be similar.
Apr
8
comment Problem on a square group
made a slight edit. Hope you don’t mind.
Apr
8
comment Proof for multiplication of two power series
One way is to take $1+u+u^2+u^3+u^4$ and square it, but look only at the terms of degree $\le4$. You should see then what’s happening, and be able to construct a proof, probably with some elements of mathematical induction in it.