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Jul
31
comment What is the group structure on the ring of power series around a point that makes it “the completion of an elliptic curve” along that point?
I think you’re making it much too hard, and also hoping for something that (I think) doesn’t exist. More via private communication.
Jul
31
comment $2$-adic sequence converging to $\sqrt{-7}$.
No, the series is not convergent when you plug in $-1$ for $t$; I didn’t say it was. But since the $n$-th degree (in $t$) term is divisible by $2^n$ after the substsitution, you definitely get a convergent series. It converges to $\sqrt{-7}$ because when you take the series in $t$ and square it, you get $1+4t$. This process is consistent with the procedure of substituting something small for $t$
Jul
29
comment 8 cubes ($2$x$2$x$2$) crossed by a straight line
@HenningMakholm, thanks, that looks right.
Jul
29
comment 8 cubes ($2$x$2$x$2$) crossed by a straight line
Well, @JamesZen, think of the three mid planes. One horizontal, one east-west, one north-south. Now your line may cross each of these mid planes, or may not. Each time it crosses a plane, it goes from one cube to another. And you see that this is the only way a line can go from one cube to another. At most three passages from one cube to another, hence at most four cubes are visited.
Jul
28
comment 8 cubes ($2$x$2$x$2$) crossed by a straight line
Probably, @Michael. The complete proof needs a verifying example.
Jul
28
comment Choice of Primitive Element in ''Primitive Element Theorem''
Oh, well, then look closely at the suggestion of @GitGud; or else just take a normal extension of $\Bbb Q$ of degree three. If it were generated by a cube root, then the cube roots of unity would be there too, no good because the cube roots of unity are quadratic over $\Bbb Q$.
Jul
28
comment Why Composition and Dihedral Group have reverse order of operation?
If my memory serves me, back when I was in grad school (an earlier geological era), there were a couple of algebra text books that used the notation $(x)f$ — “take $x$ and $f$ it”.
Jul
27
comment Unicity in $p$-adic Weierstrass Preparation Theorem: dividing both sides of an equation of formal power series
I think I’m with you on this issue. But then, I would never have proved WPrep by what seems to be Koblitz’s method.
Jul
26
comment Unicity in $p$-adic Weierstrass Preparation Theorem: dividing both sides of an equation of formal power series
It’s not convenient for me to get hold of Koblitz’s book. Could you say what Lemma 8 says?
Jul
26
comment The algebraic closure of $\mathbb{Q}_p$
Why don’t you just find an irreducible polynomial of every degree? Much simpler.
Jul
24
comment Third point of intersection is also a point of inflection?
Certainly, @GeorgesElencwajg. But nothing works till the neutral point is fixed at an inflexion point.
Jul
23
comment Choice of Primitive Element in ''Primitive Element Theorem''
You’re asking whether every extension is radical.
Jul
23
comment Neukirch's motivation for $p$-adic numbers
Well, one can argue forever on this, but I maintain that the only “natural” coefficients for $\Bbb Z_p$ are the Teichmüller representatives, which (except for $2$ and $3$) aren’t in $\Bbb Z$.
Jul
23
comment Neukirch's motivation for $p$-adic numbers
What disturbs me most about trying to give meaning to a particular ”coefficient” in a $p$-adic expansion is that these coefficients depend so strongly on your choice of representatives in $\Bbb Z$ or $\Bbb Z_p$ for the elements of $\Bbb F_p$, the residue field. For $\Bbb Z_5$, for instance, maybe you want to use $\{-2,-1,0,1,2\}$ for your possible “coefficients” of your expansion. Makes perfectly good sense. Unless you want to go all the way into abstraction and represent a $p$-adic integer by a Witt vector, it seems to me that the analogy with power series is extremely weak.
Jul
22
comment Problem on Galois theory and irreducible polynomial
OP has explained that he meant the product of $\sqrt[p]2$ and $\sqrt[q]2$ to generate the big field, so I’m not sure your analysis is completely on the mark.
Jul
22
comment Problem on Galois theory and irreducible polynomial
I’ve edited to put in a central dot. Thanks for the clarification.
Jul
22
comment Problem on Galois theory and irreducible polynomial
In lightly editing your question, I see that you have a dot between the two radicals. Did you mean a comma (virgule) or did you mean the product of the two radicals?
Jul
22
comment Problem involving cubic field extensions
Your proof in the algebraic case looks right to me, but I’m not at all sure that the statement is true in the general case. Do you have reason to think that it is?
Jul
22
comment How to evolve an expression with two denominators
I like this. Plus one point for you.
Jul
22
comment Priority of the 3 axioms of groups
I don’t think it makes sense to quote $g_3$ before $g_2$.