Mathematics - 2019-11-17 (page 2 of 2)

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Ultradark

9:01 PM

I want to take a unit square with grid lines like this:

and CHANGE the coordinate system

and see if it's homeomorphic to the original coordinate system

Actually I think I really mean CURVILINEAR coordinates

tigre

Well, I have no idea what that means, so I can't help you. Sounds like you want to endow the set $\Bbb R\times \Bbb R$ with different metrics, and compare the induced metric topology? But I'm not quite sure.

Ultradark

I want to define two different metrics in the unit square and see if there is a map between them

the first metric being the usual euclidean metric as pictured above

tigre

9:19 PM

@Ultradark You really have to work on your mathematical precision my dude

Galois in the Field

@Adam I was answering a question actually.

Ted Shifrin

@user193319 No. $\omega\wedge\eta = (-1)^{k\ell} \eta\wedge\omega$, when $\omega$ is a $k$-form or $k$-vector and $\eta$ is an $\ell$-form or $\ell$-vector.

Leaky Nun

s n i p e d

Ted Shifrin

So, @Leaky, what's your progress report for the month?

Leaky Nun

being confused by everything in Tate's thesis

Ted Shifrin

9:31 PM

well, that's not expected reading for an undergrad, anyhow.

I have never looked.

Leaky Nun

watching my home fall

Ted Shifrin

Otherwise, classes are going well? Made friends?

Yeah, your homeland is very sad. But I find almost all the world pathetic these days.

Leaky Nun

a war is happening as we speak

inside a university

we just lost the main entrance

@TedShifrin you might be able to deconfuse me of the fact that there are two different norms on $K \otimes_\Bbb Q \Bbb R$

the euclidean norm $\|(x_1, x_2, \cdots, x_{r_1+r_2})\| = \sum \|x_i\|^2$, and the field norm $\|(x_1, x_2, \cdots, x_{r_1+r_2})\| = \prod_{i=1}^{r_1} x_i \prod_{i=r_1+1}^{r_1+r_2} \|x_i\|^2$

Ted Shifrin

You're missing a square on the first one, also squares on the second?

Whoa. I have no idea what's going on now.

Leaky Nun

desmos.com/calculator/uwrxpe31uy

this is the lattice $\Bbb Z[\sqrt2]$

the purple curve is the units group

Ted Shifrin

9:39 PM

So, for the Gaussian integers you get the same norm, but in general you get different ones. I still have no idea what you're asking.

Leaky Nun

it isn't really a question

I'm just very confused as to how I'm supposed to deal with the two different norms

Ted Shifrin

The norm that shows up in Galois theory is not supposed to be a metric norm.

But you should talk to @Mathein. I'm sure he has a good answer.

When you work with real quadratic extensions, you get hyperbolas, and when you work with imaginary quadratic extensions, you get ellipses.

Leaky Nun

yeah

tigre

This is probably basic, but I'm not sure I understand what is meant by "cut out"

Suppose $A$ is a ring, and $I$ is a maximal ideal of $A$. If $f\in I$, show that the Zariski tangent space of $A/(f)$ is cut out in the Zariski tangent space of $A$ by $f\pmod {I^2}$.

Ted Shifrin

"cut out" = "defined"

tigre

9:54 PM

So the Zariski tangent space of $A/(f)$ is given by taking the maximal ideal $I'$ of $A/(f)$ corresponding to $I\subset A$ over the canonical quotient map. Then we have the Zariski tangent spaces $(I'/I'^2)^\vee$ and $(I/I^2)^\vee$, which are respectively $(A/f)/I'$ and $A/I$ vector spaces.

So the claim is that $(I'/I'^2)^\vee$ is defined in $(I/I^2)^\vee$ by $f\pmod {I^2}$?

Does it mean that we can identify it with the $(A/I)f$?

To make sense of that, I'd probably have to first realise $f$ as a dual element

Leaky Nun

10:10 PM

@TedShifrin the more I learn the more I know what I don't know

this isn't very productive towards writing my paper though

Ted Shifrin

That's a usual feeling for math people :)

Mason

11:02 PM

Is there a way to implement a cubic residue symbol in mathematica?

Cubic reciprocity

Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x3 ≡ p (mod q) is solvable if and only if x3 ≡ q (mod p) is solvable. == History == Sometime before 1748 Euler made the first conjectures about the cubic residuacity of small integers, but they were not published until 1849, after his death.Gaus...