Hey guys...I heard that if we have a signal of finite width $\Delta x$ in time then this signal has a width in the frequency domain $\Delta f$ such that $\Delta x \Delta f > c = constant$ but then again many finite signals in time spread to infinity in...anyhow I miss some points there...what is this property of fourier transform called?
we did poincaré duality, cech cohomology, alexander duality, some knot theory, homotopy groups, fibrations, cofibrations, stuff like Brown's representability theorem, obstruction theory, Postnikov systems, localization at primes and some stuff I forgot
Proof: $\operatorname{Spec}(R)$ is Noetherian, as $R$ is Noetherian and it is Hausdorff as $R$ is 0-dimensional. But, by a purely topological argument, any Noetherian Hausdorff space is finite
@TobiasKildetoft I don't know, feels more topological to me. I have not used the structure sheaf on $\operatorname{Spec}(R)$ at all, just the underlying topology
Here's a nice algebraic problem with a topological solution: let $R$ be a commutative ring with only finitely many maximal ideals and suppose that the Jacobson radical is $0$. Show that every prime ideal of $R$ is maximal.
@MatheiBoulomenos It has only one real roots and have 1+1+1+2 factorisation in Z7 as I computed, but I hope you don’t use properties that fall from the sky
@BalarkaSen Is the computation difficult without knowing that it is solvable?
@MatheiBoulomenos the response you gave me was actually false mate
from our book: "The set A has four elements: { 1, 2 }, ∅, { 1, 2, 3 }, and { 1 }. For example, { 1, 2 } ∈ A. Note that 1 is not an element of A, so 1 ∉ A, although { 1 } ∈ A. Furthermore, { 1 } ⊈ A since 1 ∉ A."
$256a^3e^3-192a^2bde^2-128a^2c^2e^2+144a^2cd^2e-27a^2d^4+144ab^2ce^2-6ab^2d^2e-80abc^2de+18abcd^3+16ac^4e-4ac^3d^2-27b^4e^2+18b^3cde-4b^3d^3-4b^2c^3e+b^2c^2d^2$ that's the discriminant of $ax^4+bx^3+cx^2+dx+e$
Just for reference: the discriminant of your polynomial is $$3125 p^4-256 N^5 p^5.$$ I computed it using Mathematica. For comparison, the discriminant of the general monic polynomial $x^5+a x^4+b x^3+c x^2+d x+e$ is $$256 a^5 e^3-192 a^4 b d e^2-128 a^4 c^2 e^2+144 a^4 c d^2 e-27 a^4 d^4+144 a^3 ...
There's a formula for the discriminant of a quintic
@MatheiBoulomenos Sorry, I forgot to think about your problem. I think it's a consequence of Chinese remainder theorem that $R \cong R/\mathfrak{m}_1 \oplus R/\mathfrak{m}_2 \oplus \cdots \oplus R/\mathfrak{m}_n$ if $\mathfrak{m}_i$ for $i = 1, \cdots, n$ are the maximal ideals of $R$?
Yes, you can solve it like that, I didn't think about that
The topological solution went like this: We show that if the Jacobson radical of $R$ is $0$, then $\operatorname{Specm}(R)$ is dense in $\operatorname{Spec}(R)$. Proof: A basis of $\operatorname{Spec}(R)$ is given by $D(x)$ where $x$ ranges over all non-nilpotent elements of $R$. Now if for some non-zero (hence non-nilpotent, as $R$ is reduced) $x$, we have $D(x) \cap \operatorname{Specm}(R) = \varnothing$, this means that $x$ is contained in every maximal ideal.
