@Adeek, the Galois group of $\mathbb{Q}(\zeta_n) \cong \mathbb{Z}_n^\times$, with $\zeta_n$ an $n$th root of unity. When $n$ is prime, $\mathbb{Z}_n^\times$ is cyclic.
by the way, does a continuous linear functional $T$ separating two subsets $E$ and $F$ of the domain Banach space with complex numbers as the codomain mean that there exists a $\gamma \in \mathbb{R}$ such that $Re (Tx) < \gamma \leq Re (Ty)$ for all $x \in E$ and $y \in F$?
@Adeek, let $n$ be prime. Then $(x^n-1)/(x-1)$ is irreducible. The Galois group of an irreducible polynomial acts transitively on its roots. Thus, for any $k$, we have an automorphism determined by $\zeta \mapsto \zeta^k$
Compose this automorphism with itself a bunch of times
Consider the maps as you said before $\omega \mapsto \omega^k$ for $k = 1,..,4$. Then Since $[Q(\omega) : \mathbb{Q}] = 4$ the above permutations of roots determine our galois group completely. Consider $\sigma_3 : \sigma_3(\omega) = \omega^3$. Then $\sigma_3^2(\omega) = (w^3)^3 = \omega^9 = \omega^2 \implies |\omega| > 2 \implies |\sigma_3| = 4.$
There are only two subgroups of order 4 either $Z_4$ or $Z_2 \times Z_2$ it can't be $Z_2 \times Z_2$ so must be $Z_4$.
@DHMO or I read it wrong. I am using a screen reader so I can't be sure. Can you look at the link where I got it from? It's on exercise 1.39 mitpress.mit.edu/sicp/chapter1/node21.html
@DHMO I am an absolute beginner on math notations. Now that you explained the correct formula, I'm just curious. Could it be the reason for the exponent 2 all over the formula in the book is because it assumes x gets updated every term? For example 3-x^2 where x is the result of x^2 from 1-x^2? Because (x*x)x? I'm not so sure, because your formula is certainly clearer. Or maybe the book's formula is really just wrong?
Suppose there is a set $X=\{a,b,c\}$ and a topology $\tau_x=\{\emptyset,\{a\},\{b\},\{a,b\},\{a,b,c\}\}=\{\emptyset,A,B,C,D\}$. From this topology and the notion of closeness given in the wikipedia link we can deduce the following: 1. no points are close to the empty set 2. $a\in A$ thus only a is close to A 3. $b \in B$ thus only b is close to B 4. $a,b, \in C$ thus both a and b are close to C 5. $a,b,c \in D$ thus all a,b,c are close to D.
Why the closure under intersections is necessary in the axiom:
Considering only real numbers in $(0,1)$ Hardy and Wright define simply normal numbers as
A number is simply normal in base $r$ ( $r >=2 $ ) if $\; \lim_{n \to \infty }\frac{n_b}{n}=\frac{1}{r}$, where $n_b$ reprsents number of occurrences of digit $b$ ( $0 \le b < r$ ) in the first $n$ digi...
However suppose I ask what is the complement of {a} in $\tau_2$, then I get {b,c} which is in $\tau_2$, thus {a} is closed. But in $\tau_2$ there is no notion of "only a is close to the set {a}" because {a} is not in the topology. Therefore, what does it mean when it is found that {a} is closed in $\tau_2$?
by "what does it mean" I mean what information can we learn from knowing that {a} is closed in the topology $\tau_2$?
Nobody said {a} must be in a topology. But based on the discussion and the notion of closeness, the open sets in a topology tell us which points or bunch of points are close/near to what sets
so if {a} is not present in the topology, then we cannot really say anything about whether a is close to the set {a} in this topology?
but from the definition of closed topological sets, we found the complement of {a} is {b,c} which is in the topology hence an open set. That means b or c are close to the set {b,c} in this topology
Therefore by definition of closed topological set, {a} is a closed set in this topology
so that implies somehow there's a notion of "a but no other points or bunch of points is close to {a}"?
Is the infinite cylinder with shrinking radius $\bigcup_{t \in \mathbb{R}} \{t\} \times \min(1,\frac{1}{|t|}) \cdot S^1$ a smooth submanifold of $\mathbb{R}^3$?
Or let's say we smooth the ugly parts at $t=1$ and $t=-1$ with a bump function.
Can the order of a differentiation and summation be interchanged, and if so, what is the basis of the justification for this?
e.g. is $\frac{\mathrm{d}}{\mathrm{d}x}\sum_{n=1}^{\infty}f_n(x)$ equal to $\sum_{n=1}^{\infty}\frac{\mathrm{d}}{\mathrm{d}x}f_n(x)$ and how can it be proven?
My intuiti...
@AliCaglayan Because we're essentially adding stuff that weren't there. But of course my intuition must be playing tricks with me. There must be some cancellation going on.
I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say
$$
\sum_{a \in A} a \quad \text{where A is some uncountable indexing set e.g. some $A \subset \mathbb{R}$ }
$$
Would it be better to avoid ...
Continuous only means the inverse image of any open set is an open set. On $\Bbb Z$ with the usual topology any function is continuous since $\{n\}$ is open
(If I understood correctly) since it is given in the usual topology that {n} is open, it means its complement is not in the topology as otherwise {n} will be closed
so {n} is open under the discrete topology, and also closed because its complement is a subset of $\mathbb{Z}$ which is open in this topology. Hence all subsets of $\mathbb{Z}$ are clopen in this topology and thus disconnected?
However suppose I ask what is the complement of {a} in $\tau_2$, then I get {b,c} which is in $\tau_2$, thus {a} is closed. But in $\tau_2$ there is no notion of "only a is close to the set {a}" because {a} is not in the topology. Therefore, what does it mean when it is found that {a} is closed in $\tau_2$?
If I understood correctly, the sets in $\tau_2$ are the open sets, which means I can say e.g. point b is close to {b}, point a or point b is close to {a,b} and points a,b,c is close to {a,b,c}. But in this topology, I don't have {a} thus knowing that {a} is closed, how I can have a notion that point a is close to {a}?
You said that closed sets contains all their limit points. So do you mean since {a} is closed in this topology, then it contains all its limit points, which is a in this case
Whereas {a,b} is open since its complement is not in the topology, thus it does not contains all its limit points (are these limit points a,b for this set?)
In fact the difference between an operation and a "loi de composition" is that the "loi de composition" transforms elements of $G$ to elements of $G$. (whereas scalar product or a norm are not "loi de composition" since they go from $E\times E\to \Bbb R$ or $E\to \Bbb R$ respectively)
$\tau_2=\{\emptyset,\{b\},\{a,b\},\{b,c\},\{a,b,c\}\}$ Let me think...
$\{b\}^C=\{a,c\}\not\in \tau_2$, thus $\{b\}$ is not closed so there is some sequence in $\{b\}$ converging to a or c, which is not in $\{b\}$.
$\{a,b\}^C=\{c\}\not\in \tau_2$, thus $\{a,b\}$ is not closed so there is some sequence in $\{a,b\}$ converging to c, which is not in $\{a,b\}$.
$\{a,b,c\}^C=\{a,b,c\}\in \tau_2$, thus $\{a,b,c\}$ is closed so there is some sequence in $\{a,b,c\}$ converging to a or b or c, which are in $\{a,b,c\}$
Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the same convergent sequences?
@secret you should also pay attention to the fact that in non metric spaces sequences are not enough to get all of the limit points, the sequential closure is usually smaller than the actual closure
(Below are my conceptual question highlights for interest of people, I am not asking them now since my brain got too tied up in knots for topology, I will try to solve them later) 1. It is clear what the intuitive notion of "two things close to each other" means in the closure operator convention of topology (which can be shown to be equivalent to the open set convention). However because the open set convention is the most widely used, I am trying to understand how the intuitive notion of "two things close to each other" is phrased in terms of the open sets in the topology.
Do you mind looking at the link again? I just want to be absolutely sure. I absolutely don't understand why my screen reading software is reading the formula wrong. It reads it as:
Considering only real numbers in $(0,1)$ Hardy and Wright define simply normal numbers as
A number is simply normal in base $r$ ( $r >=2 $ ) if $\; \lim_{n \to \infty }\frac{n_b}{n}=\frac{1}{r}$, where $n_b$ reprsents number of occurrences of digit $b$ ( $0 \le b < r$ ) in the first $n$ digi...
How to find $\lim_{n \to \infty } \frac{ \sum_{k=1}^{n}(1-(1-2^{-k})^n) }{\ln n}$.Is there a simple way?
Can the order of a differentiation and summation be interchanged, and if so, what is the basis of the justification for this?
e.g. is $\frac{\mathrm{d}}{\mathrm{d}x}\sum_{n=1}^{\infty}f_n(x)$ equal to $\sum_{n=1}^{\infty}\frac{\mathrm{d}}{\mathrm{d}x}f_n(x)$ and how can it be proven?
My intuiti...
Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the same convergent sequences?
