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22h
comment Techniques to prove that two field extensions are distinct
Yes, your objection is definitely well-taken. Would you like to e-mail me, and we can continue this discussion that way? Find my e-mail on my page, and the link is on my profile.
1d
comment Techniques to prove that two field extensions are distinct
Yes, I assumed that much of this would go over your head, which is why I gave the “method” first, and then its justification. I was thinking of @JyrkiLahtonen when I prepared this. But the method should always be useful: find a prime $p$ with three special properties: $\widetilde f\ne\widetilde g$; $\widetilde f$ and $\widetilde g$ are both “unramified”, by which I mean that each has no repeated roots (over $\Bbb F_p$, you recall); and the two of these are not only unequal, but the degrees of their splitting into irreducibles over $\Bbb F_p$ induce different partitions of the degree $n$.
2d
answered Techniques to prove that two field extensions are distinct
Apr
24
comment Techniques to prove that two field extensions are distinct
I’m still away, as soon as I get home I’ll answer this properly. My comment above is right enough, but a bit beside the point. As opposed to looking at the very hard-to-evaluate discriminants, a much better way to tell that two number fields are nonisomorphic is to look at the splitting of primes in the two fields. For these two fields, one has a prime of degree one above $3$, the other does not.
Apr
23
comment Techniques to prove that two field extensions are distinct
The fields are different because the corresponding extensions of $\Bbb Q_3$ are already different. Sorry, don’t have time to explain further.
Apr
23
comment Prove that $\log_a(1/x)=-\log(x)$.
Logarithm is not defined for zero and negative numbers anyway.
Apr
23
answered Extension of Completions of Number Fields
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
13
awarded  Generalist
Apr
13
answered Is the algebraic closure of $\mathbb Q$ in the field $\mathbb Q_5$ a number field?
Apr
12
answered A doubt about real analysis concerning countable sets
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
answered Finite extnsion of local field.
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.