Mathematics

user131753

3:05 AM

Does anyone know whether the classifying space functor from the category of groups to the category of topological space is a full functor?

Rithaniel

How does one find the norm for a particular algebraic number field? Like $\mathbb{Z}[\sqrt[3]{-2},\sqrt{-2}]$ as an example.

(It always seems to happen. Two or three people with questions show up at the same time)

Silent

Let $f(x) = (x^ 2 − 2)(x^ 2 − 3)(x^ 2 − 6)$. For every prime number p, show that $f(x) ≡ 0 \pmod p$ has a solution in $\Bbb Z$.

I have no idea what to do

Rithaniel

That sounds like a job for quadratic reciprocity, Silent.

At least I'm pretty sure. Just show that 2, 3, or 6 is a quadratic residue mod p for each prime p.

Silent

@Rithaniel I have encountered term 'quadratic residue' first time, so will you please explain a little bit more?

lio

Hi everybody, I've posted a question here (https://math.stackexchange.com/q/3203462/662117),
could you please check it for any help

Rithaniel

3:19 AM

Gotcha. So $x$ is a quadratic residue mod $p$ where $p$ is prime if there exists an $a$ such that $a^2\equiv x$ mod $p$.

You generally show a number is a quadratic residue via use of the Legendre symbol (if I've spelled that correctly). Which has it's own fairly simple (but detailed) arithmetic associated with it.

The Legendre symbol $\binom{x}{p}$ is 1 if x is a quadratic residue mod p, or -1 if it's not. If $x=a_1a_2\cdots a_n$ then $\binom{x}{p}=\binom{a_1}{p}\binom{a_2}{p}\cdots\binom{a_n}{p}$

(Typing this out on mobile is difficult)

There are other details, but that should be enough to solve this particular problem, I hope.

@Silent (In case you'd like a ping)

Silent

Thank you so much, Rithaniel, for going through all this pain.

Rithaniel

3:37 AM

Ah, it's okay. Had some time to kill. :P

lio

guys, take a few seconds to check my question, I lack a good maths base to be sure of my approach there

Eiffelbear

3:55 AM

hello guys

can a vector, not a scalar, follow a probability distribution?

Silent

4:20 AM

@Rithaniel, so even $x^2-2$, $x^-3$ and $x^2-6$ each is individually zero mod p for any prime mod p right?

user8718165

4:57 AM

Hello can somebody please help me with this question?

0

Q: Question about definite integrals

Please assume that this graph is a highly magnified section of the derivative of some function, say $F(x)$. Let's denote the derivative by $f(x)$. Let's denote the width of a sample by $h$ where $$h\rightarrow0$$ Now, for finding the area under the curve between the bounds $a ~\& ~b $ we can a...

Secret

6:07 AM

@Ultradark You can try doing a finite difference to get rid of the sum and then compare term by term. Otherwise I am terrible at anything to do with primes that I don't know the identities of $\pi (n)$ well

Secret

6:22 AM

in The h Bar, 43 secs ago, by Secret

That is, given a language L where each word has a semantic meaning, is it possible to find some string s such that s cannot be ascribed semantic meaning for any extensions L' of L

Leaky Nun

7:40 AM

@TedShifrin @Semiclassical mathcounterexamples.net/…

Zee

What an ugly function

Odestheory12

8:07 AM

if $F' = f$, is true that $F' \circ g = f \circ g$?

Leaky Nun

@Odestheory12 yes

Rithaniel

9:57 AM

@Silent No, take for example the prime 3. 2 is not a residue mod 3, so there is no $x\in\mathbb{Z}$ such that $x^2-2\equiv 0$ mod $3$.

However, you have two cases to consider. The first where $\binom{2}{p}=-1$ and $\binom{3}{p}=-1$ (In which case what does $\binom{6}{p}$ equal?) and the case where one or the other of $\binom{2}{p}$ and $\binom{3}{p}$ equals 1.

Also, probably something useful for congruence, if you didn't already know: If $a_1\equiv b_1\text{mod}(p)$ and $a_2\equiv b_2\text{mod}(p)$, then $a_1a_2\equiv b_1b_2\text{mod}(p)$

BAYMAX

10:21 AM

just a number theory question

remainder when $5^103$ divided by 14

I am trying to apply the congruence but getting no trcks

ÍgjøgnumMeg

$5^{103}$?

BAYMAX

oh

yes

sorry

$5^{103}$ is divided by 14

ÍgjøgnumMeg

Divide $103$ by $14$ with remainder, use Euler's theorem

BAYMAX

oh cool!

got the remainder as 5

cheers!

ÍgjøgnumMeg

10:52 AM

:)

Glator

11:14 AM

is Q(π) simple extension of Q ?

ÍgjøgnumMeg

yes

A simple transcendental extension

Glator

11:32 AM

@ÍgjøgnumMeg is it necessary in definition of simple extension of F that the element which we are going to adjoin to F is root of some polynomial of F ?

Alessandro Codenotti

No, just an extension by a single element

So it is either isomorphic to an extension obtained adding the root of a polynomial or to F(X)

tarit goswami

2:04 PM

Is there any book or article that explains the motivations of the definitions of group, ring , field, ideal etc. of abstract algebra and/or gives a geometric or visual representation to Galois theory ?

Secret

Abstract Algebra: A Geometric Approach

ÍgjøgnumMeg

@Ted plug

Secret

what I am not sure though is how large the galois theory section is

tarit goswami

2:24 PM

@Secret Thanks :)