serious-discussion

[timid bronze]

What is this

[solar hawk]

The best soup in the world, it doesn’t taste like egg at all

It’s made with the stuff inside eggs tho

[deep mango]

Oh hell no

I had egg drop soup recently

it's so gelatinous

[vocal vessel]

Wonton soup > egg drop soup

[solar hawk]

Ew wh

Bro that prolly just went bad..

Mine is always soupy

[spice osprey]

Hiii

Someone pls help me with my proj hahahaha I did the design I jst need help with all every math parts

[sonic field]

i hate hate HATE zuchini pasta. BUT. miracle noodle started makin these noodles out of egg whites, probably gonna use those 🔥

[deep mango]

Ew

[sonic field]

ok fair

i thought that too until i tasted their spaghetti and vermicelli. it actually do taste like noodles, far better than anything i've tried so far

[deep mango]

Also made with egg white?

[sonic field]

yeh

[timid bronze]

Why are you a hater….

[sonic field]

it was these

i was shocked

[sonic field]

i jus don like the crunchy texture and mine for w/e reason always comes out with a lot fo water which ruins the pasta

[deep mango]

Huh

crazy

[sonic field]

that vermicelli hit that ramen itch SO hard

[hexed tangle]

https://tenor.com/view/peaky-blinders-gif-19842754

[timid bronze]

Not bad looking

[timid bronze]

#❓how-to-get-help

[timid bronze]

You don’t know how to do it right.

[sonic field]

i knew you were gonna say that

[deep mango]

if its so easy to do wrong then that's something wrong with it

[sonic field]

someone stuck in the sushi-with-bbq-sauce era

[deep mango]

i agree with jay but i like the crunch so i like it anyway

[magic python]

:3

[timid bronze]

Not necessarily

[sonic field]

i just can't get over that crunch when i'm tryin to eat spaghetti 🤣

[timid bronze]

Would you there’s something wrong with a roux

A roux is pretty easy to mess up

You just have skill issues ryc the bird

[deep mango]

"hey guys mac and cheese isnt really all that"

"JUST MAKE A ROUX BRO"

so annoying

nothing wrong with roux but if a recipe says "make a roux" then im less likely to make that recipe

[timid bronze]

Ok but my point is that some things are just hard to perfect

[modest matrix]

[timid bronze]

And I’m confident Jay is not perfecting the noodles.

[timid bronze]

They are right though

I don’t like mac and cheese that much but a roux does elevate it

Though imo as long as the mac and cheese is homemade and seasoned I’d eat it

Unseasoned mac and cheese is so gross

[deep mango]

seasoned with what...

it's called "mac" and "cheese"

that's 2 things

[timid bronze]

Get your boring self out of here

@velvet dagger This lil freak ryc eats unseasoned mac and cheese

[deep mango]

what would you even put in it

chicken?

[timid bronze]

Paprika, cajun, garlic powder, chili powder, etc

[timid bronze]

Mf what

[deep mango]

Who wants their mac and cheese spicy

omg this makes so much more sense

[turbid sedge]

Hi fiscussy 2

[timid bronze]

Definitely not your unflavorful palette

☠️

Hi Lich

[turbid sedge]

Ay all what's up

[timid bronze]

Nothing much

What about you

[deep mango]

Reddit is fucking insane

1-2 inches THICK potato chips!?

[turbid sedge]

Js had dinner

[turbid sedge]

I guess...

r/iminsane

[sonic field]

atthat point they not even chips anymore 😔

[timid bronze]

What was it

[timid bronze]

I saw people put potato chips in their brownies

[deep mango]

I saw a potato chip cookie recently

im curious because i love potato chips on my sandwiches

[timid bronze]

I hate potato chips but they work in sandwiches ngl

[deep mango]

ok maybe this subreddit is based

[timid bronze]

Wht

[white lichen]

what's the difference between that and the rest of reddit

[timid bronze]

Ok but ew

[solar hawk]

Yay

[turbid sedge]

Ay it was rice and soya beans

[sonic field]

me while they arguin about food

[turbid sedge]

Good catch

[timid bronze]

Who is arguing

[snow ice]

time to remake react os but for windows 11

also who can help me get rid of this clover next to me

[solar hawk]

React os??

[snow ice]

yes

[solar hawk]

😭please lord tell me this is not related to reactJS in any way

[snow ice]

not its not related

[solar hawk]

Oh ok

LMAO

For a second

[snow ice]

its literally just an os that's like windows but made from scratch

[solar hawk]

Why is it called react

[snow ice]

as a combination of "Reaction" to Microsoft's OS monopoly and "Operating System" (OS). Do you get it?

[old oak]

What's wrong with that?

I mean, other than the "bro"

[snow ice]

what is a "ROUX"?

[magic python]

French cheese sauce iirc

[snow ice]

why are we talking about culinary dishes in here rn

it must be because of ally and ryc.

should I ping them

[old oak]

A French chef: https://en.wikipedia.org/wiki/Michel_Roux

[snow ice]

I see...

[old oak]

Not cheese, and not really sauce; it's flour cooked in fat, and it's used as a thickening agent: https://en.wikipedia.org/wiki/Roux

If you add milk to a roux you get bechamel, which is a sauce

And if can add cheese to a bechamel, which people certainly do

[magic python]

Ohhhhhh I didn't know roux was the name of the flour thickener, so cheese sauce is made USING roux, but isn't roux?

[old oak]

https://en.wikipedia.org/wiki/Mornay_sauce

[old oak]

Yeah, it's basically roux -> bechamel (by adding milk) -> mornay (by adding cheese)

[snow ice]

odd idea, so this is a type of old, non-synthetic agent to thicken and puff up products (like cake) made with just flour and fat?

[magic python]

Oooo I love all the culinary terms that we have

[old oak]

It doesn't puff them up, just thicken them, it's mostly used in various sauces, stews and sometimes maybe soups.

[snow ice]

oh, as in making liquids more viscuous?

[old oak]

Pretty much

[snow ice]

pretty neat, though per two table spoons its around 120 calories.

[magic python]

That's how fat works

[old oak]

Well, you will also use a roux to make a savory souffle, which is the one non-liquid example I can think of

[magic python]

Although you don't use all too much of it

[old oak]

It does start as a liquid until baked, of course

[snow ice]

I guess so, doesn't cornstarch do something similar too?

[old oak]

Speak for yourself! https://en.wikipedia.org/wiki/Deep-fried_butter

[magic python]

Oh, instant heart attack, right up there with instant spoiled milk in spells

[old oak]

One guy on Top Chef made deep-fried mayonnaise once

[magic python]

He lost right

[snow ice]

who is crazy enough to do that

[magic python]

Like immediately

[snow ice]

this is literally the most american thing you could have said

[old oak]

Nope, it was a Paula Deen challenge so he either won or placed high.

[magic python]

I feel bad for whoever had to taste it

[old oak]

Molecular gastronomy and doing crazy things with food was sort of his thing

For the fondue challenge he did one where you dipped things in liquid nitrogen instead of molten cheese or hot fat

[snow ice]

so is this competition just making the quirkiest food possible?

[old oak]

Not really, it's about making the best food possible, but his thing was getting quirky with it while still being good.

He almost won his original season and did win the all-stars

[snow ice]

I see, then that's goofy and also epic aswell.

[snow ice]

wow.

[serene vault]

https://tenor.com/view/fox-gif-24544794

[old oak]

Delicious

[snow ice]

the owl or the fox?

[old oak]

I don't think it's good practice to contemplate the deliciousness of fellow server users

[snow ice]

you gave me another idea now

[quasi jetty]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[snow ice]

what...

[old oak]

This adds to my already long list of regrets

[serene vault]

https://tenor.com/view/fox-foxes-cute-snow-poke-gif-9290384749482771871

[snow ice]

cool, just put this one in font size 24 and make it bold+italicized and in yellow underline, thx

[snow ice]

ooh, do I eat the fox, the hand, or the snow

[serene vault]

whichever gives you the least regret

[snow ice]

oh ok, I'll just eat the snow

[ornate sierra]

wsp nerd homies

anyone needs help?

[snow ice]

no thanks

go to the help forums

they might have people

[ornate sierra]

im asking others you are not included g

[serene vault]

there's the forum and occupied channels if you're looking for ppl to help

[ornate sierra]

did i ask?

[serene vault]

¯_(ツ)_/¯

[ornate sierra]

do u think i am cool?

do you have braincells?

[snow ice]

definitely yes.

[snow ice]

and maybe, idk you at all

[ornate sierra]

which means no

[ornate sierra]

are u a nerd?

[snow ice]

???

[snow ice]

I am a nerd

[ornate sierra]

same sadly

🙁

[snow ice]

what do you know about math?

[ornate sierra]

like algebra

algebra@

[magic python]

Which type, there are multiple

[ornate sierra]

any

[magic python]

What's a field.

[ornate sierra]

you wanna know the definition or explanation?

[magic python]

Definition I suppose, as that is what's taught for the algebra iirc

[ornate sierra]

like field is a number system where u can do all the normal arithmetic youre used to

like add subtract divide

according to mathematics

[magic python]

Okay... what is a groupoid (from the same math field, algebra)

[ornate sierra]

like its a collection of objects where som pairs can be combined (like doing an operation) but not necessarily all of em)

[magic python]

It's closed, no?

[ornate sierra]

ye a groupoid is closed but only on the pairs of elements where the operation is defined

[snow ice]

what must groupoids have?

[magic python]

Can you give a slightly more formal definition of a groupoid M?

[ornate sierra]

like are u tryna be more specific? with the answer

[snow ice]

yes?

[ornate sierra]

wait my wifi router was turnedd of sorry

[snow ice]

...

[ornate sierra]

hold on let me type

[magic python]

I want how you'd define the operation the groupoid is equipped with

[ornate sierra]

A groupoid must have:

Objects

Arrows between objects

Composition (only when arrows match)

Identity arrow for each object

Inverse for every arrow

[magic python]

No

[ornate sierra]

then?

[magic python]

That is not a groupoid

[ornate sierra]

it is a groupoid

[magic python]

It is certainly one, but that's not the definition

That is much too strict

[ornate sierra]

i already typed the definiton earlier..

if u want another definition i will type it for u

[magic python]

Define it's operation.

[ornate sierra]

like a groupoid operation is a partially defined binary operation ∘
:
𝐺
×
𝐺
→
𝐺

such that

the product g.h is defined if the target of h equals the source of g

[snow ice]

dude this is pasted from chatgpt

!nogpt

[quasi jetty]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[magic python]

Well how come you are using a separate operation in the definition than the one you are defining?

[ornate sierra]

hello?

excuse me?

[snow ice]

chatgpt when you paste from it makes a bunch of enters between operations

and you literally changed the font of the letters

[magic python]

You used a strange circle to be the operation to define, but then you used ×

Also earlier I said a groupoid M.

Purely so it would be harder for an LLM to tell you, as you would need to put that info in

[placid shadow]

Doakes!!!

[ornate sierra]

ye??

[placid shadow]

https://tenor.com/view/dexter-gif-1878760417600646737

My favorite character in Dexter

[ornate sierra]

S T F U !

[placid shadow]

o

[ornate sierra]

STOP TRYNA BE ME BITCH ASS MFS

[placid shadow]

I'm not! dont worry

[ornate sierra]

aight

[magic python]

Also I'm asking for a groupoid, not a partial groupoid like your first definition was.

[placid shadow]

👍

[ornate sierra]

me chill

alright hold on car

can i ask u a question?

[white lichen]

is everything after s8 good

[snow ice]

hello @placid shadow !

[placid shadow]

Yeeeee, it's pretty great if you liked the previous stuff!

[magic python]

Ig

[white lichen]

I've put off the last 4 eps of s8 forever because of how bad they seem to be 💔💔

[placid shadow]

Noooooooooooooooooooooooooooooo those were so good!!!

You have no idea

You need to watch them!!! if you want

[ornate sierra]

alright homie car the question is if a number is increased by 7, and the result becomes 25.
What was the original number?

[white lichen]

well maybe I'll like them cuz I watch any movie or series and think it's peak

[white lichen]

i will hopefully soon enough

[magic python]

18 assuming we are using standard definitions taught in elementary algebra

[white lichen]

I'll sleep now, good night

[snow ice]

its 18

[ornate sierra]

wrong

[placid shadow]

Goodnight sleep tight

[snow ice]

then do tell

[magic python]

I said assuming those definitions.

Not whatever random version of subtraction or addition you want

[ornate sierra]

since we added 7 twice and subtracted only once, we must subtract 7 again

so x = 18-7 = 11

so the answer is 11

[magic python]

That's not using the standard definition of increased by.

[snow ice]

we increased by 7

where did we subtract

[ornate sierra]

wait im wrong lol

the answer is 18😭

[magic python]

Also you literally could've lied and looked on wikipedia to do this, but no, you gave me a partial groupoid

Wikipedia has a definition

[ornate sierra]

how tf it was a partial groupoid can u elaborate?

[ornate sierra]

why would i cheat?

[snow ice]

you did alr?

[magic python]

A partial groupoid is when its not necessarily all of them, a groupoid needs it to be all of them

[ornate sierra]

oh

[magic python]

Well you are misunderstanding what I'm asking at the very least

[ornate sierra]

maybe

can u explain me whats a gropuoid and partial groupoid

[snow ice]

a groupoid should have an identity aswell

and an inverse

[ornate sierra]

guys i gtg eat food

we will meet again

[snow ice]

wait it doesn't need to?

[ornate sierra]

soon

[magic python]

Although I don't technically have enough proof it's an LLM, as you may just switch fonts and format weird

[ornate sierra]

bye guys

[magic python]

A groupoid is a set M that has a binary operation that closes it (M²->M)

[serene vault]

man I just got to get half-comfy with centralizers and shit and now I have to deal with the offspring of a group?

[magic python]

A partial groupoid is one where it can be undefined (0/0 type of stuff) and such, it's only a partial binary operation

[snow ice]

isn't that like a magma

oh wait

[magic python]

Nope! Although I should probably use its more common name, a magma, I used groupoid because an LLM may mistake it for cat theory iirc

[snow ice]

ur talking about universal algebra

[magic python]

Abstract algebra yes

[snow ice]

then its a groupoid

[magic python]

As they boasted about how they know "any" type of algebra

[snow ice]

a groupoid in abstract algebra is just any set with a binary operation

[magic python]

A closed one

Same as a magma

They are different terms for the same object

[magic python]

I should've looked up a harder field named algebra but I wouldn't know enough about it to know if it was correct or not

[true zinc]

A groupoid is also a small category where all.morphisms are isomorphisms (I think)

[true zinc]

Commutative algebra has some very silly monsters in it

[junior veldt]

yes

[magic python]

Wowzers is that still algebra or is that cat theory

[junior veldt]

not necessarily small

[magic python]

Ooooo

[junior veldt]

probably cat theory

[grand wraith]

What symbols on keyboard and mouse represent a fraction?

[ocean harbor]

I'm not sure what do you mean by this

but fraction is /

[idle steppe]

Whats the definition of a binary operation in group theory? Im just a highschooler

What is and isnt a binary operation?

For one i guess you would want the binary operation to always give the same, unique result when u do the same thing

So, 2+2 should always be equal to 4

But what else?

[clever harbor]

operation that takes in two things and outputs one

[clever harbor]

"well-defined" is the term for this

you want it to behave nicely with groups, so to formalise: if a = b and c = d, then a + c = b + d (where + is your operation)

and where a,b,c,d are elements of the group

[idle steppe]

I would totally self study groups if i wasnt already busy with self studying physics

[clever harbor]

it can truly be whatever, but interesting operations are often commutative (a + c = c + a) or associative ( (a + b) + c = a + (b + c) )

[clever harbor]

so many fun topics, so little time

[idle steppe]

Its wild we've moved on from using letters instead of numbers to using a universal symbol instead of an operation

[clever harbor]

yeah ikr

if you end up doing math in uni, it gets so abstract

[idle steppe]

Earlier i was convinced that i'd do maths but now im concived i'll do physics

[clever harbor]

some schools have math-phys programs

best of both worlds

[idle steppe]

Im pretty sure i would have to move abroad to do such a program

[clever harbor]

dang

they are very complimentary fields, major in one minor in the other?

[idle steppe]

I would try something like that if the uni i intend to go to, has it

Theres like 2 unis in my country that have a maths or a physics program

[junior veldt]

It’s a function from G x G to G

[timid bronze]

@solar hawk

I want to see you play this Roblox game

@jagged forge

My friend sent me a pic of it

😭

[sonic field]

amukh would probably die 🤣

it reminds me of the nextbot on garry's mod

what's the name...

MUNCI

https://www.youtube.com/shorts/1tXS_nrzM_g

[timid bronze]

LMAO

I was not expecting that shit to come closer ☠️

[sonic field]

@hexed tangle ?? oh wait mb

[shadow palm]

Same here LMAO

[solar hawk]

LMAO

[solar hawk]

Dude I’m lit playing a Roblox server horror game

Roadside shawarma

[timid bronze]

I thought you that you don’t like horror games

Well.. W in that case

[snow ice]

how tf does this happen on roblox 😭

wth is this game

[dreamy oracle]

if u have 31.999
how woulud u round it to the nearest hundreth
my brain is not working rn

[snow ice]

32

[timid bronze]

@deep mango

[deep mango]

what

[timid bronze]

What's with the attitude

[deep mango]

sick, tired, grumpy, but go ahead. take a chance

[timid bronze]

Oh um

Nvm then

Feel better

[deep mango]

hmph

did you want to play a game

i just got home after a long day so i think i will rest

[snow ice]

good day

there we go I added another prefix to the front of my name

[old oak]

Still a virus, though

[snow ice]

not anymore!

dis means not

un means not

and anti means not

-1*-1*-1=-1

[old oak]

Actually -1-1-1 = -3

[snow ice]

stoopid astrixes

by discord

-1⋅-1⋅-1=-1

[old oak]

Also "unvirus" doesn't really work because "un" doesn't go before nouns.

It's a verb-modifying prefix

[snow ice]

I guess so

[old oak]

Adjective as well, possibly

[snow ice]

but adding prefixes is fun

[old oak]

Unforgivable

[latent edge]

co-cohomology

[old oak]

Sure, why not

[snow ice]

that works

I'll add that next week

or maybe "re"

[old oak]

I can't wait

It promises to be hugely droll

[snow ice]

@silent junco is scaring me, they are watching me...

[silent junco]

uh

[old oak]

It is inadvisable to leave children unsupervised

[snow ice]

:(

jk ur fine

[silent junco]

i was just thinking about how to phrase my attention seeking message tbh i wasnt paying attention to you

[snow ice]

ah ok

mb

[silent junco]

no worries

[snow ice]

:)

[snow ice]

what are you trying to say btw?

[silent junco]

oh the moment has passed dont worry about it

[snow ice]

... ok

[latent edge]

@snow ice remember the group theory discussion we had

[snow ice]

YES

[silent root]

Hey what is the best entry to become an option trader in mathematic ways? /s

[latent edge]

Okay here's a fun problem to think about

[snow ice]

mmhmm

[latent edge]

Last time we came up with 3 groups with the same size

[snow ice]

yep

[latent edge]

And all of them were isomorphic to each other

[snow ice]

yep

[latent edge]

Okay if I give you any group G with size 2 once again

What can you say about it?

[snow ice]

its albenian

[latent edge]

yes abelian

[vast wraith]

https://tenor.com/view/albania-heart-locket-love-gif-10159499668458955942

[latent edge]

what else

[snow ice]

cyclic?

[latent edge]

can you list me the elements?

[latent edge]

Why?

[snow ice]

{0, 1}

[latent edge]

what is 0 here

[snow ice]

e if we let G = {e, a}?

[latent edge]

Yes I want this

When you write {0,1} you are implicitly choosing an explicit group

I just want to think about this abstractly

[snow ice]

i see

[latent edge]

{e,a} with any operation

[snow ice]

now, to answer your question of why G is cyclic

e is the identity, and since a is obviously not e

a*a=e

[latent edge]

wdym by this

[snow ice]

oh wait

its with any operation right

so we can literally say EVERYTHING about this

[latent edge]

yes we can take * as a notation it's fine

[snow ice]

alright

[latent edge]

The point is which you kinda made is the only things you can do is do things with e

and a*a = e

[snow ice]

yes

[latent edge]

yes and by this we can see this "abstract" group of size 2 is isomorphic to the 3 groups we had yesterday

[snow ice]

yep

wait I'm rly sorry but I gtg :(. I'll come back shortly dw.

:(

[latent edge]

This tells you there is only one group of size 2 up to isomorphism

[torpid bay]

alright

[vast wraith]

yeah this is the crux of the argument, but you have to explain why a*a=e

[timid bronze]

@sonic field @jagged forge Ok this HAS to be trolling https://www.tiktok.com/t/ZThd8e9HL/

[snow ice]

first, due to closure, the only two possibilities are a⋅a=a or a⋅a=e.
we can then disprove why a⋅a=a
multiply both sides by a^-1
a⋅a^-1⋅a=a⋅a^-1
a⋅a^-1 is the same thing as the identity of the group, e, as we want to keep it abstract.-->
a=e, which is proven false under the group G with set {e, a}, so a⋅a cannot = a and furthermore must equal e

[vast wraith]

lovely

[snow ice]

:D

math is pretty

also, to add on, if we go as in a⋅a^-1=1-->
a=1
for e to be the identity of this group, it must also be 1, but now its not a distinct element anymore as a=e, and is proven false under the group G with set {e, a}

[ocean robin]

abstract algebra mentioned

[snow ice]

we love abstract algebra

[ocean robin]

yez

[turbid sedge]

I know my ip adress

No you don't

😡

[snow ice]

also appearentally google considers tung tung tung sahur an artist

[turbid sedge]

WHAT

[snow ice]

😭

search it up

[turbid sedge]

triple T got his recognition bru

[snow ice]

LOL

[ocean harbor]

its finally here

[snow ice]

:)

[ocean harbor]

i didnt listen to f-777 last year 🥀

[snow ice]

:|

[ocean harbor]

do u know who that is

[jagged forge]

why do i have to install the app to watch tiktok videos 🥀

[silent junco]

https://www.tnktok.com/t/ZThd8e9HL/

Holly (@shepherdplayspiano)

❤️ 10.7K 💬 744 🔁 3.4K

▶ Play video

fxTikTok · Jan 2, 2026, 2:01 AM

wtf

there we go

oh it still doesnt work

[timid bronze]

🥀

[silent junco]

finally

[silent junco]

@jagged forge

[timid bronze]

Nothing wrong with install TikTok suremark

You might find an egirl through there

[silent junco]

i wouldnt wish that on my worst enemy

[timid bronze]

LMFAO

Shhhh I’m just trying to piss him off

[silent junco]

🥀

[jagged forge]

uhm. did she just say enhance the flavor with water

[silent junco]

idk i didnt watch it

[jagged forge]

i’m in agony

i have a bookshelf but nothing to put on it

my lamp and statue are too big

[silent junco]

ask ryc for bookstore recs

[jagged forge]

i was thinking i’d go by that math bookstore again

[silent junco]

which one

the strand?

[jagged forge]

yeah that one

[silent junco]

yeah good collection

i got two books from there

[jagged forge]

nice

[snow ice]

yes, the dj who has more famously made music for geometry dash

[latent edge]

Was expecting to see Elliptic Curves

I'm still not disappointed

peak

[timid bronze]

Tbf I’d do anything to maintain clear skin

[jagged forge]

it’s like when you see cucumber water being sold for $8 at the store

[vast wraith]

i love soups

[snow ice]

I'd do anything to keep people away from hearing my mic

[timid bronze]

Skincare is fun but a lot of the skincare tips online piss me off

Like wym putting lemon on your skin is good

[snow ice]

it's "natural"

[vast wraith]

it’s a crazy world out there

[timid bronze]

😭

[snow ice]

why the heck do we need this bro

[timid bronze]

Can’t believe there’s also people against sunscreen

[snow ice]

sunscreen is goated

[timid bronze]

Ngl my entire family is against sunscreen

I’m the only one in the family who wears it

[snow ice]

how should I help these innocent 9 year olds from sending laughing discos in the chat

[timid bronze]

😭

[timid bronze]

What

[snow ice]

there are literal youtubers doing looped live streams begging for laughing disco emojis to be gifted

each one costs around 10$ to gift

and kids are genuinely falling for it

[vast wraith]

ok enough internet for me tonight bye

[timid bronze]

Btw what in the AI 😭

[snow ice]

someone made $10k off of one of them

[limber thunder]

someone made like 400k in polymarket off the Venezuela thing today

insider trading moment

[snow ice]

bro I have this product, 5 star, I use my computer heat to make it work

[latent edge]

wait wrong channel lmao

[timid bronze]

Ngl my old MacBook could give anyone a 3rd degree burn

[snow ice]

you petrify me every day

[timid bronze]

If you put it on your lap for too long

[snow ice]

u use mac?

[timid bronze]

Yes I am a proud full time mac user

[snow ice]

cool, I use window

[latent edge]

I use arch btw

[limber thunder]

the name "laptop" is kinda misplaced why would you want a 90°C processor anywhere near youtr lap

[snow ice]

I should have expected you use arch linux

[latent edge]

see totally arch

[ocean harbor]

real

[latent edge]

w*ndows

[snow ice]

I use react os

[latent edge]

you need to seek help if you really do

[snow ice]

react os is 10x better than windows

[timid bronze]

I mean my newer Mac’s fan don’t go crazy

My old Mac would be an airplane after opening up a single Google tab

[snow ice]

ITS FREE WINDOWS 95.

[latent edge]

Mac is alright

I would recommend someone to buy a mac rather than a windows laptop any day

[limber thunder]

that's kind of a low bar to compare with

[snow ice]

fair point

😭

[ocean harbor]

notability?

[timid bronze]

Yes

[ocean harbor]

nice

[timid bronze]

I used to be a GoodNotes princess but

I realize how bad it is compared to Notability now

[ocean harbor]

goodnotes sucks ass

I use note+

[timid bronze]

Fr

[snow ice]

I use notepad++ for my notes lol

[timid bronze]

Haven’t heard of Note+

How is it

[limber thunder]

based and redpilled

[ocean harbor]

want me to show you a sample? its my favourite

[limber thunder]

my one friend in the pandemic (i had to do online classes)

[snow ice]

notepad++ is great

[timid bronze]

Sure hehe

[ocean harbor]

i like everything except for chatgpt

[timid bronze]

😭

[limber thunder]

yeah tbh it's pretty great software considering it's open source and all

decent tablet support

lots of features

[snow ice]

u can write code on it too

[limber thunder]

hmm I've never tried this

[snow ice]

no other notepad without being paid can do that

[timid bronze]

It pisses me off that these note taking apps are obsessed AI

[limber thunder]

or do you mean like the LaTeX support

the LaTeX support is acc pretty great

[timid bronze]

But I’m in love with Notability’s audio recording

[snow ice]

no like it's its own client like vscode, it recognizes the language

[timid bronze]

So… I won’t give up the app

Plus I like the infinite amount of templates

[limber thunder]

[snow ice]

though latex is also great too

[latent edge]

it fucking sucks like shit lags for centuries just to get an AI response which only makes you want to throw your device cuz the output was horseshit

[snow ice]

notepad++ has no ai in it

[ocean harbor]

ai pen is useless

[latent edge]

ai pen my ass

[ocean harbor]

the rest are good

[limber thunder]

it's pre-AI

[timid bronze]

Your notes are so pretty

[ocean harbor]

thanks

[latent edge]

These notes are lovely btw

[snow ice]

ig but the latest update was dec 27 2025, they def could have added ai in it

[dry copper]

Those notes are beautiful

[timid bronze]

What is an AI pen…..

[snow ice]

what's a linux?

[ocean harbor]

it generates u stuff when u snap something also its paid

I find it really stupid at first

[snow ice]

,tex So p2w notes, I just use the math discord texit bot from now on

[fathom swallow]

Disunantivirus

[ocean harbor]

also with this u can link any websites and use them like an extension

I linked wolfram alpha and desmos on it

[timid bronze]

What the fuck…? Anyways..

(Hi! Thanks for letting me borrow your profile picture @velvet dagger )

[snow ice]

wth is this

[ocean harbor]

LMFAOOOOOOO

[latent edge]

btw akira how do you find the composure to write good notes

Like I just get bored and go to exercises immediately and start scribbling stuff

[timid bronze]

Meet Damietta ✨

[ocean harbor]

its fun to write because I like organizing stuff

there was a class I shouldn't have done notes for it bc i got D on it

[latent edge]

Like even if I write notes it's very rare to look at them again

[ocean harbor]

and I killed my own wrist

[limber thunder]

eh what for

[ocean harbor]

yeah I can relate

[limber thunder]

OCR ig

[timid bronze]

It’s hard to write neat without a grid background

[limber thunder]

i got an e-reader and it's kinda awkward to write in it

[timid bronze]

I use grid background then change it back to blank after I’m done

[limber thunder]

also my handwriting is shit

[timid bronze]

Mine too 😭

[latent edge]

Latexing my notes is cool but gosh it takes so long I just give up at some point

[timid bronze]

My handwriting on exams is also horrible as fuck

[latent edge]

i thought windows was doing the absolute cinema thing

https://tenor.com/view/absolute-cinema-absolute-cinema-femrover-rover-gif-14890473560168326644

[latent edge]

idek what was going on here im ngl

[ocean harbor]

i understand nothing here

[latent edge]

same

[limber thunder]

generational change

[limber thunder]

tf is going on there

[latent edge]

idk

[limber thunder]

I see like a short seq (???) and a determinant

I'm not alg pilled

[latent edge]

I think that was during the research period

I recognize the ugly ass sums

[torpid bay]

the margins could fit that large sum, but unfortunately not the trace of a 2by2 matrix

rip

[ocean harbor]

cannot enter discussion 1 anymore too many blocked messages lol

[latent edge]

mood

[ocean harbor]

nice handwriting

[latent edge]

This one gives me butterflies in my stomach

[timid bronze]

It’s dogshit writing.

But tbf that’s when I was a GoodNites user

Laggy ass app

[latent edge]

peak

[timid bronze]

Praying Notability treats me better

[ocean harbor]

what do u think of mine

[latent edge]

i dont have my uni notes in this pc lmao

[torpid bay]

oh

dirichlwet hcars

[timid bronze]

Cutesy notes

[latent edge]

yeah

[torpid bay]

that one actually looks like i might be able to understand like a few sentences of it then hm

[latent edge]

you should understand this

it's literally what we discussed last time

I even wrote these when I was texting u

[torpid bay]

oh dam ye

[timid bronze]

People who can write xi perfectly are my ops

[latent edge]

well expect the algebraic geometry slop above

[torpid bay]

i didn't know dirichlet chars connect to galois stuffs, looks like a lot to look forward to

[latent edge]

oh yeah they do

I initially didn't know that too

I just learned them for L functions

kinda shame ngl cuz the story is really nice

i can explain if u are interested

[torpid bay]

I was doing stuff with generalizing zeta function and got a thing where I could represent it either with J(n)/n^x, as (1-J(p)p^-x)^-1 or as (J(n)u(n)/n^x)^-1, where J(n) is a multiplicative function, so i was wondering why go for the more specific dirichlet chars

which have the extra condition of modulo stuff

[latent edge]

I see

[torpid bay]

(but ye explain the other thging after dis as well)

[latent edge]

well one way to motivate these things is proving the classical theorem on arithmetic progression on prime numbers by Dirichlet

[torpid bay]

is that the thing that says given a ax+b where a and b are coprime or smthn, then there are infinite primes of said form?

[foggy meadow]

I wish I knew more about this type of stuff.

[timid bronze]

Btw does the app you use have audio recording

[latent edge]

yeah it says if gcd(a,q) = q then there are infinitely many primes p = a mod q

[ocean harbor]

yes

[latent edge]

so the problem is you want multiplicative stuff to deal with these things

so that you can use analytic stuff

[torpid bay]

[latent edge]

what dirichlet characters offered is the orthogonality behaviour they have

$\frac{1}{\phi(q)} \sum_{\chi} \bar{\chi(a)} \chi(n)$

[fathom swallow]

nDeltoid

[latent edge]

this guy equals 1 if n = a mod q and 0 otherwise

[toxic dagger]

Isn't there a simpler way to prove it

[torpid bay]

ah condition 2

[latent edge]

yeah condition 2 is useful

anyways the thing I wrote acts as an indicator function

for n = a mod q

it is multiplicative too for free

[velvet dagger]

Damietta is CRAZY WORK

[timid bronze]

Apparently it’s a place in Egypt

I just googled

[latent edge]

yeah from there you do similar things you do for the case of just regular primes

the only difference here is you have to worry about characters in your analysis

okay that's the analytic story

Let's go algebraic now

Ok so are you familiar with roots of unity?

[torpid bay]

i love em

[latent edge]

hell yeah

You can take an extension over Q where you attach roots of unity

These are called cyclotomic fields

They are SUPER nice

[torpid bay]

I love the cyclotomic polynomials as well, so its nice to see some love for cyclotomic

[latent edge]

To give some examples say you look zeta_4

what is zeta_4

btw zeta_n = exp(2pi i/n)

[torpid bay]

1,i,-i,-1 then?

[latent edge]

right yeah

what would Q(i)/Q be?

well 1 and -1 are already in Q

[hexed tangle]

[latent edge]

the only new thing here is i and -i but these differ by a -1 so really the only new thing we are attaching is i

[hexed tangle]

deltoid!

[torpid bay]

is Q(i) the extension of the rationals by i, and then Q(i)/Q im not sure about the / notaiton

[latent edge]

I just put /Q to emphasis we are looking at extensions over Q

[torpid bay]

ok

[latent edge]

yeah so these are just elements of the form a+bi where a,b in Q right

okay nice

now if you view this as a vector space over Q

what basis can you have?

What would be the dimension of this vector space

[torpid bay]

2

[latent edge]

right

okay how familiar are you with Galois groups?

[torpid bay]

I've only heard they were used for the polynomials insolvability of deg 5 and above or smthn, other than that nothin

[latent edge]

Can you compute the Galois group of this extension?

If not that's fine we can go over this

[torpid bay]

no

[latent edge]

okay so not every extension is Galois

so im swiping some details here under the rug

but if you believe me this extension is Galois so we can work with it

okay so elements of Gal(Q(i)/Q) are going to be automorphisms that fix every rational number and permute roots

the rule of thumb here is that the number of elements is equal to the degree of the extension

so here we have that the degree is 2

so we must have 2 elements

one obvious one is the identity element

what it does is it just sends i -> i

the other one flips the sign

so it does this i -> -i

another way to see this is considering the minimal polynomial

[torpid bay]

x^2+1

[latent edge]

yes exactly

now set this equal to zero

then apply sigma in Gal(Q(i)/Q)

[torpid bay]

they cancel each other out right?

[latent edge]

show me what you have

[torpid bay]

its just i + -i = 0

[latent edge]

no I meant apply sigma to the polynomial

sigma(x^2+1)

[torpid bay]

sigma of x^1+ over the gal set?

right

[latent edge]

yeah

[torpid bay]

1^2+1 + (-1)^2+1 = 4

wait a minute

nvm

[latent edge]

now sigma is a field automorphism so you can split this guy up like this sigma(x^2+1) = sigma(x^2) + sigma(1)

then remember I said sigma fixes elements over Q and here 1 is such an element so sigma(1)=1

also you can raise the exponent from inside to outside

so overall you have this sigma(x)^2 = -1

which you can think of as y^2 = -1

now it's clear what values of y should we take

[torpid bay]

ye, the i and -i ones

[latent edge]

yup exactly

so overall the group has elements {1,sigma} where sigma here is the flipping sign one

@torpid bay so far so good?

I feel like I lost u : (

https://tenor.com/view/fumo-fumofumo-reimu-reimu-fumo-marisa-gif-16432408177096756319

[torpid bay]

"now sigma is a field automorphism so you can..."

[torpid bay]

I'm not really sure why a sum would be an automorphism?

[latent edge]

here it just means you can do this sigma(x+y) = sigma(x)+sigma(y) and sigma(xy)=sigma(x)sigma(y)

[torpid bay]

$\sum_{Gal(Q(1)/Q)}{x^2+1}=\sum_{{1,-1}}{x^2+1}$ this is what i have written down

[fathom swallow]

Yeatte

[latent edge]

oh no it's a function

the elements of the Galois group are functions

[torpid bay]

im trying to scroll up to see what i missed

[latent edge]

sigma(x^2+1) = sigma(x^2)+sigma(1) = sigma(x^2)+1 = sigma(x)^2 + 1

sigma(x)^2 + 1 = 0

[flat orbit]

Not functioning

[pure hollow]

Galois group?? Cosmic Galois groups.

[latent edge]

No.

[pure hollow]

Sad

[torpid bay]

hm, what does it mean to fix the base field?

[latent edge]

it means sigma(a) = a for every a in the base field

alright are we clear now?

[torpid bay]

so Galois group of Q(i)/Q is the set of automorphisms for Q(i)/Q? and that would be isomoprihc to the set {1,-1} that act on which Q(i)/Q?

[latent edge]

it's the set of automorphisms of Q(i) that fix Q

the set would be {1, tau} where 1 is the identity automorphism that sends i -> i and tau: i->-i

also known as complex conjugation

so if you are given an element a+bi in Q(i)/Q, applying the identity would do nothing

But tau(a+bi) = a-bi

[torpid bay]

[latent edge]

coolio

and you know this is isomorphic to Z/2Z right

[torpid bay]

yes

[latent edge]

but this is not quite useful yet

Now we can see what happens in general for zeta_n

you are just looking at z^n = 1 right

[torpid bay]

ye

[latent edge]

okay are you familiar with the solutions of this polynomial over C?

[torpid bay]

z^n=1 can be factored into precisly the cyclotomic polynomials phi_k(x) where k and n share a prime factor

and there are iirc euler totient(k) solutiosn to the k'th coylocotmic polynomial

also mobius in there as well

[latent edge]

right yeah

the thing to focus on here is the primitive ones

These guys will be in the Galois group of Q(zeta_n)/Q

namely the elements are going to send zeta_n -> zeta^k_n

and since we are talking primitive

there is phi(n) of them as you noted

So the Galois group is isomorphic to a group of order phi(n)

Do you know any such group

Hint: Dirichlet characters

[torpid bay]

i feel like prime numbers may pop up here, but the only groups i know are like C_2 and C_3. so im guessing something to do with finite simple groups relating to rotations of polygons?

[timid bronze]

How I feel when studying math. None of it makes sense!

[latent edge]

Kinda close yeah

[ocean harbor]

what about physics

[latent edge]

It is (Z/nZ)^x

The integers modulo n under multiplication

[timid bronze]

Physics makes sense but that shit pisses me TF off cz it's so easy to get confused

[latent edge]

this group by definition has phi(n) elements

[latent edge]

can you make sense of this pls thank you

anyways

okay so all and all

We have that Gal(Q(zeta_n)/Q) = (Z/nZ)^x

https://tenor.com/view/absolute-cinema-absolute-cinema-femrover-rover-gif-14890473560168326644

[timid bronze]

Oops I pissed off a math major.

[timid bronze]

Ok but physicists do piss me off with their notation

[ocean harbor]

true

[latent edge]

@torpid bay You might see where is this going now right

[torpid bay]

ye dirichlet modulo

[latent edge]

Dirichlet characters are characters chi: (Z/nZ)^x -> C^x

but we also have the isomorphism with the Galois group

So you can think of them as characters of Gal(Q(zeta_n)/Q)

[torpid bay]

a little question, that x is the degree thing we calculated earlier, right? the degree of the minimal polynomial iirc

[pure hollow]

Relatable

[latent edge]

deg of min poly = degree of extension = deg of galois group

wait no i said something stupid

[torpid bay]

[latent edge]

no that's right nvm lol

[torpid bay]

[latent edge]

the min poly of the cyclotomic extension is the cyclotomic polynomial right

[snow ice]

we have the best notations

[torpid bay]

yes

[latent edge]

yeah alright

[timid bronze]

No

[latent edge]

you might ask if there is a 1-1 correspodence between characters and fields

[timid bronze]

Chemists and physicists piss me off with notations

[latent edge]

Just like in Galois theory you have subfields <-> subgroups

[torpid bay]

mhm

[snow ice]

SPLIT UP DF/DX

[latent edge]

you can do that

you take a character and find the kernel then look at the fixed field formed by the kernel

For example let's take Q(zeta_8)

Gal(Q(zeta_8)/Q) = (Z/8Z)^x which has phi(8) = 4 elements

right

{1,3,5,7}

[torpid bay]

ye

[timid bronze]

Mathematicians aren't annoying with notations except idk why some of them get angry when I use prime notation instead of leibniz notation.

[latent edge]

alright consider this character for example chi(1), chi(3)=-1, chi(5) = -1, chi(7)=1

[snow ice]

I use newton notation

[timid bronze]

The hell is Newton notation

[latent edge]

what is the kernel of this character?

i.e things that map to 1

[torpid bay]

1,7

[latent edge]

right

now under the isomorphism what does this correspond to?

[timid bronze]

Ok nvm this is ragebait

[latent edge]

peak zeta function btw

[pure hollow]

THIS IS SO TRUE

[latent edge]

despite the unfortunate naming

[hexed tangle]

indeed mate

[snow ice]

its easier to write bro

[timid bronze]

Nah

Euler and Newton notation is ragebait

[hexed tangle]

ally is such a panda

[pure hollow]

Alhfor?

[snow ice]

its easier to write newton notation

[timid bronze]

Wut does that mean

[snow ice]

it is quicker

[hexed tangle]

yes

[pure hollow]

Euler is simply peak

[hexed tangle]

i mean like you're adorable and cute like a panda ofc

[snow ice]

considering how much work I have to show for all of my problems

[timid bronze]

;-;

[snow ice]

ok wtf

[hexed tangle]

what

[latent edge]

btw just to recap the isomorphism is this k -> (sigma_k: zeta_n -> zeta^k_n)
@torpid bay

[snow ice]

this sounds so odd

[hexed tangle]

blud

[pure hollow]

I get newton but Euler is good it's unambiguous

[snow ice]

where's the blood?

[latent edge]

ugh this is so confusing lmao

[torpid bay]

k?

[hexed tangle]

get your head out of the gutter

[latent edge]

yeah k in (Z/nZ)^x

[timid bronze]

You def saying this because I'm unemployed 🤣

[latent edge]

so in our case we would have sigma_1 and sigma_k

[hexed tangle]

oh you got it!

[timid bronze]

FUCK YOU

[torpid bay]

${\sum}_1$?

[timid bronze]

I'M IGNORING you

[viral yew]

unemployed or underemployed better

[fathom swallow]

Yeatte

[torpid bay]

or like below it?

[hexed tangle]

pwease i'm sorry

[latent edge]

no sigma as in the elements in the Galois group

[snow ice]

layed off

[latent edge]

the functions

[viral yew]

that happened to my pringles can

[hexed tangle]

https://tenor.com/view/panda-huahua-gif-1476795351732195837

[latent edge]

for each k in (Z/nZ)^x you attach a function sigma_k

[torpid bay]

oh

[latent edge]

and that sigma_k raises the nth root of unity to the k power

[torpid bay]

yeah i've been writing sigma as in sum i was confused

[latent edge]

yeah no worries

[torpid bay]

i sohld prob write the other one lel

[timid bronze]

This one TikTok video still fries me

[latent edge]

$\sigma_k(q)$

[fathom swallow]

nDeltoid

[latent edge]

anyways so the fixed field is whatever sigma_1 and sigma_7 is fixing

also we need the degree

[viral yew]

😢 💀

[ocean harbor]

lmfao

[latent edge]

this is given by phi(n)/|kernel|

so in this case phi(8)/2 = 4/2 = 2

so similar to Q(i) it's going to be of degree 2

now from there it's just playing around with roots of unity to pinpoint exactly which one

But I made it easier for you

sigma_7 is complex conjugation

[latent edge]

this means the field is real

because elements are fixed under conjugation

[torpid bay]

[latent edge]

another way to do this is you want to find a linear combination of zeta^k_n's

such that doing sigma_k doesn't change it

once you find it then you literally just compute what that sum is

[snow ice]

gemini said hmmm in their own message 😭 . the ai was thinking mid response😭

[latent edge]

this will give you the root

let me see if I can find this

okay that's fine

[viral yew]

good boy

[latent edge]

take zeta_8+zeta^7_8

[timid bronze]

It was the MC zombie sound that took me out

[latent edge]

,w exp(2pi i/8)+exp(2pi i/8)^7

[viral yew]

grhhhhhhhh

[latent edge]

mamma mia

[fathom swallow]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by despairful_deltoid

[latent edge]

okay so it is sqrt(2)

this is also real

so the answer is Q(sqrt(2))

[torpid bay]

$Q(\zeta_8)\rightarrow Q(\sqrt{2})$

[fathom swallow]

Yeatte

[latent edge]

are you trying to draw the diagram

you can btw change the chi's and play around and see what else you get. In this case all of them are going to be of degree 2 just because of how things are set up. In our case we set chi(7)=1

you could set chi(5)=1

or chi(3)=1

[torpid bay]

you set 1,3,5,7 to 1,-1,-1,1, why set 2 of them to 1 and not the others?

[latent edge]

well you want to satisfy the orthogonality relationship

if this doesn't make you happy you can do this way: we know dirichlet characters are group homomorphisms so whatever generates (Z/nZ)^x should also generate things in the characters

by checking you can see that 3 and 5 I think are generators of (Z/8Z)^x = {1,3,5,7}

[torpid bay]

ye

[latent edge]

right now we take characters and wonder what sort of things we should get

so since chi(3)^2 = 1, chi(3) can be either 1 or -1

[torpid bay]

ye

[latent edge]

same thing with chi(5)

now you just use multiplicativity and try to generate 4 different characters

[torpid bay]

3*3 = 9 = 1, so chi(3)^2 = chi(1), so chi(1) = 1 always, chi(5)^2 = 25 = 1 same thing, chi(7)^2 = 1, chi(3)chi(5)=chi(15) = chi(7)

ye

[latent edge]

yeah good job

[torpid bay]

so long as the numbers ive given to dem chis satisfies chi props, i can use those to work with the sigmas?

[latent edge]

it's obvious here we are going to have 4 cases right
(chi(3),chi(5)) = (1,1),(1,-1),(-1,1),(-1,-1)