Mathematics - 2023-07-23

冥王 Hades

00:02

@GratefulDisciple Many. Rent-A-Girlfriend being the most recent

copper.hat

02:30

definitely edgy

user10478

02:51

Is there any sort of search engine or database in which you can enter an arbitrary Cayley Table and it will return all known phenomena that Cayley Table models?

(when I say arbitrary Cayley Table, I mean preferably the input doesn't have to satisfy all group axioms)

one potato two potato

05:21

6 stars for 'about 25'!

2

user4539917

06:18

7 actually, your comment has 6 :P

123

09:09

Hello Everyone

porridgemathematics

10:25

Why is it necessarily true that in a circular planar region, which is just an open disk with finitely many open disks removed, that if a piecewise smooth curve joins points of the boundary of the region on different boundary components , but otherwise stays inside the region, and does not disconnect the region, that the connectivity of the region decreases exactly by one after cutting the region along the curve?

I can see this is true if the curve were a Jordan arc

but im not sure how to prove it for a general piecewise smooth cut

even in the case of a Jordan arc, I can only see why its true, but I cant furnish a formal proof (im guessing we could contract away the Jordan arc, and this would be sufficient to show the connectivity decreases by one)

Sorry, a circular region has finitely many closed disks removed (its open after all)

Vitalie Ghelbert

11:42

Good morning, even if it is evening. I using chat room for first time. Please forgive if I do something wrong. Can I ask here about one problem of logic? Thank You.

$ \pi $ Latex?

S is False.
https://chat.openai.com/share/c9ba0560-73a7-4c8c-96b6-1ccfdee60476

user 85795

11:59

Why not copy and paste the question in here @VitalieGhelbert?

Vitalie Ghelbert

How?

Just as simple copy and paste, just so?

user 85795

Yup

Vitalie Ghelbert

Problem of Logic:

Argument A imply statement S is true, otherwise S is false.
Argument B is equivalent with A.
If we accept B, A become non-reason argument to imply S is true, therefore S is false.
If we reject B, we reject A that imply S is true, resulting S is false.

Is S true of false?

Why I am asking, because ChatGPT deduced a solution and I was wondering because the answer is what I expected.

S is False.
https://chat.openai.com/share/c9ba0560-73a7-4c8c-96b6-1ccfdee60476

porridgemathematics

12:17

what is a non reason argument, why does accepting B make S false? Why does rejecting B imply S is false? The entire premise makes no sense to me

seems like you just wrote If we accept B, ... , therefore S is false

then you wrote If we reject B , ... , S is false

so of course chat gpt is going to say S is false based on your premises

because you just wrote S is false

Vitalie Ghelbert

No, i asked the question "Is S true of false?", taking in account A, B relations with statement S.

porridgemathematics

yeah, but what do you mean by If we accept B ... S is false, If we reject B ... S is false

you wrote some stuff in the ... part, which makes no sense to me

honestly it seems like it doesnt matter what you wrote in the ... part

when you combine those two premises, you get S is false

if I was to translate what you wrote , it would be: A <-> S , A <-> B , B -> !S, !B -> !S

Vitalie Ghelbert

I just formulated relations between arguments A, B and statement S.
After he answered the question, I give him the values for A, B and S.
And He once again returned the answer I was expecting!
That is way I was so amazed! How can that be!
He resolved a Logica Problem.

porridgemathematics

lol

chatgpt is just very polite

Vitalie Ghelbert

:) Yes. It is impossible. Because all Internet says contrary to this solution we get!
One against infinitely many! :D War!

porridgemathematics

12:27

itll give you an answer to anything at all, including nonsensical questions

what would be interesting is if chatgpt could tell you that your premises are contradictory

maybe you should ask it

Vitalie Ghelbert

Yes. When I asked Him the question "is 0 odd or even?" I get the classical answer "0 is even", that we find everywhere on the internet. But when I give Him reformulated Problem of Logic, he concluded what I was expecting! Statement "0 is even" is false! One against infinitely many on the internet! War! :D

porridgemathematics

or rather, your premises arent contradictory, but they should result in A (and B) being false

okay, but when you start putting concrete things into your premises, you cant expect a valid conclusion

ugh, it doesnt matter

Vitalie Ghelbert

Maybe. But that is what resulted from first shot of question. ChatCPT follow just rationament of implications, therefore we get what we expected.

porridgemathematics

you were expecting 0 is even to be false based on?

Vitalie Ghelbert

I was expecting he will conclude "0 is even" to be false. And He did that. :)

porridgemathematics

12:36

okay, so what?

doesnt that show chatgpt is pretty unrefined?

Vitalie Ghelbert

That show that ChatGPT can be learn to answer as a parrot, but if to reformulate the concrete logical problem in logical term, He will reveal the TRUTH, no matter what! :D

porridgemathematics

maybe im misunderstanding something, if this is a sign that chatgpt can be gamed, how does that do anything but discredit chatgpt or this type of LLM as a means to be useful? It shows that the validity of the model is totally dependent on the way you frame questions you ask it

and it cant detect when certain frames are not universal

like yours

Vitalie Ghelbert

ChatGPT is kind of soldier. You give him Theory of War, he learn all of these and answering just as a parrot. But when we give to Him logical problem formulated to get the result, he reveal the truth, no matter what! :D

porridgemathematics

are you unironically saying this? its hard for me to tell

im going to assume you are being facetious

Vitalie Ghelbert

He is good of learning what You give to Him as a good student.
But if You want to reveal the truth, we can ask him reformulated Problem of Logic.
And he follow logical implications and relationship between arguments and statements. :)

porridgemathematics

12:42

im not so sure you arent an LLM...

Vitalie Ghelbert

No, I am not LLM. I just could not swallow all of the resources from the internet saying "0 is even".
Therefore experimented with ChatGPT to see what solution will He return.
He returned what I was expecting: statement "0 is even" is false.

porridgemathematics

interesting, ive not heard of using chatgpt for emotional validation before

Vitalie Ghelbert

I am sending to Him 🌹and He is answering very polite. Often I forget that He is AI. :)

PlaceReporter99

13:17

@VitalieGhelbert 0 is actually even.

@VitalieGhelbert that’s not good. Just remember to remember that it’s AI.

Vitalie Ghelbert

Please forgive my daring, but I do consider "zero parity is neutral", and demonstrating with a + 0 = a.

porridgemathematics

hes been scarred accepting the whole internet points to zero being even, and in response has resorted to gaslighting chatgpt to tell him otherwise

which is obviously what you do when the entire internet points out you are wrong about something

that or he is trolling

Vitalie Ghelbert

I cannot trust some information, only because 7 billion people on the planet Earth believe so. :D

porridgemathematics

thats the spirit

Vitalie Ghelbert

Trust, but verify. Ask Yourself first: why so? 🤔

copper.hat

13:27

Another delete after getting an answer question, I suspect homework math.stackexchange.com/q/4740824/27978

PlaceReporter99

14:34

@VitalieGhelbert what is $\frac02$, to start off?

I just found out you can do ---O--- to make something that looks like theta: O

Vitalie Ghelbert

Is the definition we use n/2 for |n| > 0, where we select odd from even numbers, but when n = 0, 0/2 = 0 cannot be unique argument to consider statement S "0 is even" true.

PlaceReporter99

@VitalieGhelbert but if $\frac x2 \equiv 0 \pmod 2$ then $x$ is even.

Vitalie Ghelbert

Yes, for |n| > 0, definition using n/2 = (q, r) works just and fine.
The anomaly is when we have n = 0.
0/2 = 0 is not anymore a valid argument to prove statement "0 is even" as true.

shintuku

what's your definition of an even number?

Vitalie Ghelbert

As usual, n/2 = (q, r), but considering only |n| > 0.
I am considering "0 parity is neutral", for n = 0, resulting a + 0 = a.

shintuku

14:50

what's q and r?

Vitalie Ghelbert

q is quotient, count number of groups of 2 elements to consider number even.
r is remainder, count number of elements of incomplete groups of 2 elements to consider number odd.

shintuku

oh ok, so a number is even if the remainder is 0, except if the number being divided is 0?

Vitalie Ghelbert

Examples:
n/2 = (q, r)
4/2 = (2, 0)
5/2 = (2, 1)

Exactly.

shintuku

well yeah that works but it's not a standard definition

Vitalie Ghelbert

Yes, but we can come to such anomaly.
How many dance couples can we count if we have 10 boys and 10 girls?
How many dance couples can we count if we have 10 boys and 9 girls?
How many dance couples can we count if we have 0 boys and 0 girls?

Resulting in the last question, 0 couples is even? :D

shintuku

15:03

yeah that works, but as i said it's not the standard definition

Vitalie Ghelbert

Yes. But I am asking to answer not for others, but for me.
Because what I am thinking is what I have. :)

shintuku

sure, but keep in mind that you need to specify your definition of evenness if you're trying to tell them 0 is even, they will not understand you at first or will think you misunderstood the standard definition of even

e.g. you will confuse people if you say: "chatgpt correctly concluded 0 is not even"

Vitalie Ghelbert

I do not trying either to convince anyone else what I am using for me.

I am using this proposition for me.

Proposition:
Zero parity is neutral.

Demonstration:
a + 0 = a

That is what I am using just for me.
Like hamburger I get in my hand only.

But I can share with who is asking:
- please, give a peace of what You have for You?
- Yes, please, be welcome. Enjoy. :)

Ted Shifrin

15:39

Even+even = even; odd + even = odd. Thus, $0$ is even. Try again.

shintuku

zero parity is neutral is an axiom that creates a new parity, it doesn't admit demonstration @VitalieGhelbert

Vitalie Ghelbert

Please forgive daring of my joke;

can we buy even number of coffees less than 2?

:p Please do not mind.

shintuku

that depends on your definition of even!

again it's all fine that you want to define evenness to specifically avoid 0, but this is a definitional issue, you can't demonstrate it

Vitalie Ghelbert

:D

— 1 is even, 3 is odd, 7 is even ...
— Hey! What is that?!
— My conventions of my only "mathematics".

:p

shintuku

well yeah, again, you can do whatever you like and define whatever you like in any way, and if it creates a consistent system, you can develop a system of deduction for it

it's just not gonna be standard

Vitalie Ghelbert

15:49

:D

Yes, we can, if supposing that convention is an agreement because of convenience,
that does not follow "logical deductive reasoning",
convention that follow like "because I want that",
not because "it is behaving like that".

:)

shintuku

"convention" here means commonly agreed logical axioms

it does, in fact, function on that basis

now your axioms can be inconsistent and incomplete, and you can study these properties, and you're free to do so

Vitalie Ghelbert

:)

Yes, yes. More than that.

Statements can be "proved or disproved" to be true or false.
Statements that were proved true are used as true conventions named Theorems,
to prove other statements to "prove or disprove" and so on.

That is what were resulted in Mathematics.

:)

shintuku

no, the convention does not arise at the level of proving

the convention arises at the level of axioms

a theorem is not a convention, the axioms that produce the theorem are a convention

so if you want to modify an axiom s.t. 0 is excluded from being even, you're free to do so

Vitalie Ghelbert

*

I do understand that Axioms are statements that are true used as conventions named Axioms,
to "prove or disprove" other statements.

That is how i understand.
My how to.

*

shintuku

nothing else is happening other than breaking convention

Vitalie Ghelbert

15:54

*

Alphabets are conventions.
Digits are conventions.

*

*

Images on coins are conventions of two civilizations that were far away in time,
without exchanging information between them,
but they used "logical deductive reasoning" to count those coins same way.

I think so. Could be?

*

shintuku

you are misunderstand what I mean by deductive reasoning

Formal system

A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system".In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined system of abstract thought. The term formalism is sometimes a rough synonym for formal system, but it also refers to a given style of notation, for example...

Xander Henderson

What's with all the asterisks?

shintuku

gosh how do i prevent the wikipedia article from expanding?

Xander Henderson

@shintuku Add more text to the comment?

Vitalie Ghelbert

Sorry. I put them to make more space.

Xander Henderson

15:59

If all you provide is a link, you'll get a onebox.

@VitalieGhelbert Please don't.

Vitalie Ghelbert

O'key. I wont.

shintuku

@XanderHenderson thanks!

Vitalie Ghelbert

Let me please to entertain You with my attempt to Squaring the Circle Problem.
Pj5lf.png (1920×1200)
https://i.sstatic.net/Pj5lf.png

Have fun. :)

Xander Henderson

No, thank you.

In order to construct a circle and square with the same area, you need to be able to construct $\pi$, which can't be done. This is generally proved when one studies Galois theory.

Vitalie Ghelbert

Yes. That is what I am thinking in my free time.

Absolutely correct.

But I asked myself: it it will be, then how will it be?

If we use classical pi as is, how can we construct square and circle same area?

Can we no matter how, construct at least square and circle same area,
avoiding rules of only ruler and compass.

At least ... :)

shintuku

16:12

yeah using calculus this is pretty easy, the ruler and compass restriction is what makes it difficult

Vitalie Ghelbert

I did it even simple!
If use "at least ..." principle,
and I just wanted to see: "if it will be, then how it will be" principle,
i used only value of 3 instead of pi.

Resulted this beauty! https://i.sstatic.net/BnG0L.png

Xander Henderson

@VitalieGhelbert Sure. If you change the rules, you can do new things. The problem is not "Is it possible to make a circle and square with the same area?" The problem is "Using only a compass and straightedge, is it possible to make a circle and a square with the same area?"

It's like playing a game. Golf, for example.

冥王 Hades

Can’t square a circle using a compass and straightedge

can’t cube a square either

Xander Henderson

You are walking up to a tee and saying "Hey, I can get the ball in the hole without taking a single stroke! You just pick up the ball and carry it over to the hole. Easy peasy!"

And sure, you can do that. But you aren't playing golf anymore.

(Which sounds great to me, 'cause golf is stupid, but that's not the point.)

Vitalie Ghelbert

Yes, but I did it as is, using just 3 value instead of pi.
It is like: i do not want to die to see what is in heaven, i just want to look, how is there?

Xander Henderson

16:21

But... $\pi$ isn't $3$. If you declare that $\pi = 3$, you can prove anything you like.

Vitalie Ghelbert

True. pi is not equal 3, but pi = 3 + (tail of nail) :D

If compare 3 with tail of pi, it is like to refuse to eat apple, because I have no spoon. :D

shintuku

i move to ban aphorisms from the chatroom

aphorisms are in fact a sort of meme, so this is perfectly consistent

Vitalie Ghelbert

Please, i mine mind aphorisms sounded like metaphors. This word these days is often meet everywhere.

Xander Henderson

@VitalieGhelbert The things that you are saying make absolutely no sense.

Vitalie Ghelbert

Some time I am using word etymology to read about them.
Do You to?

Xander Henderson

16:26

You are not communicating.

Which, perhaps, is your aim.

Vitalie Ghelbert

I is more probably that You have right,
because I know from my experience,
I often do things wrong.

shintuku

ban aphorisms, quick!

Vitalie Ghelbert

etymonline.com/word/aphorism

Interesting word. I learned something new today: aphorism. Hm. Nice word and meaning.

Thank You.

shintuku

np

Vitalie Ghelbert

@XanderHenderson Thank You. 🌹My first test of reply. 0123 ...

copper.hat

17:05

How many undeletes does an question need to get undeleted? I am guessing that this was a homework question that got deleted when answered math.stackexchange.com/q/4740824/27978

geocalc33

Any geometric analysts in the house?

Xander Henderson

@copper.hat Three, I believe.

(though this number scales up as the number of upvotes on the question and answers increases)

robjohn

@冥王Hades Squaring a Circle

3

copper.hat

Thanks @XanderHenderson

Xander Henderson

@robjohn Yes, but can you cube a circle?

robjohn

17:12

I need to transcend into the fourth dimension.

Xander Henderson

@robjohn But it's only a three dimensional manifold!

robjohn

@XanderHenderson Yeah, but viewed in $\mathbb{R}^3$ it gets pretty blotchy.

Xander Henderson

@robjohn Why would you bother embedding it? er... submersion-ing it?

Never trust an umbrella.

They're pretty shady.

geocalc33

umbrellas always break down in the rain

copper.hat

@geocalc33 how do you know that the don't break down otherwise? how often do you used them when it is not raining :-)

Xander Henderson

17:24

@copper.hat I live in Arizona. I basically only use an umbrella when it isn't raining.

Gotta keep the sun off.

geocalc33

Exactly, they always break down in the rain

copper.hat

@XanderHenderson instagram.com/reel/CuM1It9ot7j/?igshid=MTc4MmM1YmI2Ng==

Xander Henderson

@copper.hat Heh.

robjohn

They never opened the cans

Xander Henderson

@robjohn It was too hot.

Vitalie Ghelbert

17:29

:)

Clever mathematical joke question and answer:
— Can You please tell me what will be "f(x) = ?"?
— Mm, f(x) = y.
— Correct.

:D

copper.hat

i remember a few years ago i brought my kids to Ireland where there was a heat wave (it was hotter than Chicago when we passed through), a few days later folks were complaining about the heat on the talk shows.

Soumik Mukherjee

17:46

Why is it showing "page not found" if I click on the mse profile of Vitalie Ghelbert?

copper.hat

i have seen that before with some other users.

Jakobian

@Thorgott hey I'm trying to learn transcendental bases and I'm going through a proof that makes transcendence degree well-defined but I'm a little stuck on it

Let $E$ be a subfield of $F$, $T$ and $T'$ are transcendence bases over $E$ and $|T'|\leq |T|$ with $|T|$ infinite

For every $\alpha\in T'$ choose finite $T_\alpha\subseteq T$ such that $\alpha$ is algebraic over $E(T_\alpha)$

Now I want to choose $\beta\in T\setminus\bigcup_{\alpha\in T'} T_\alpha$ and obtain a contradiction

How do I do this?

geocalc33

18:03

motion to remove the messages sent at 13:29

copper.hat

what timezone :-)

geocalc33

although I have told my fair share of poor jokes so maybe I shouldnt talk

@copper.hat EST

copper.hat

:-)

geocalc33

If you had to listen to one song 500 times on repeat which would you select/

I would listen to the shortest song on spotify

robjohn

@SoumikMukherjee this one?

s.harp

18:17

@geocalc33 reddit.com/r/LinkinPark/comments/8dmtho/…

Shaun

Hi :) Quick question: can Hackenbush be played on a non-planar graph? Do the surreal numbers change?

By the way, I'm on a train for the next few hours.

Hopefully, you know what I mean by the last question . . .

Xander Henderson

@Shaun Nope. No idea.

Shaun

I mean: if you can't play the usual Hackenbush games on a non-planar graph, then do you get numbers other than surreal numbers if you try?

I'm thinking of a 3D version.

Jakobian

Is anyone else familiar with field theory here?

I'm not changing my topic of study but this seems to be important for some topology things

Soumik Mukherjee

@robjohn Yes, but in the chat the rep is showing $1529$, why so?

robjohn

18:29

@SoumikMukherjee because that is the total rep in all the sites.

Soumik Mukherjee

Oh okay

Jakobian

By the way I was looking at the proof in Stacks project

But I don't really get how $E(T')$ is algebraic over $E(\bigcup_{\alpha\in T'}T_\alpha)$

I don't have the knowledge to deduce some transitivity property like this

I'm trying to do this directly, probably not the best approach.

All I really know is what it means to be algebraically independent over a subfield

And I guess, I never seen what it means for a field to be algebraic over another field

Just what it means for an element to be algebraic over a subfield

So I don't fully understand the proof there

shintuku

@Jakobian what's $T_\alpha$

Jakobian

A finite subset of $T$

shintuku

definition of $x$ algebraic over field $F$ is just satisfaction of polynomial with coefficients in $F$

Jakobian

18:37

You can scroll up to see what I wrote previously

shintuku

$\mathbb C$ algebraic over $\mathbb R$

Jakobian

I mean. Can you please read what I already wrote because you're explaining things to me that I already said I know

shintuku

@Jakobian what's the statement

transcendence bases have same cardinality?

Jakobian

Yes

shintuku

well there's no contradiction if the transcendence basis here is infinite, no?

maybe you wanted finitely generated field extension?

Jakobian

18:48

@shintuku wdym

@shintuku no

There needs to be contradiction with me taking $\beta$

shintuku

what would be the contradiction?

冥王 Hades

I wanna eat a nothingburger

Jakobian

@shintuku wdym

shintuku

i don't understand the proof strategy

Jakobian

Well, I want to show $T = \bigcup_{\alpha\in T'} T_\alpha$

Soumik Mukherjee

19:02

@Jakobian I guess it meant a field is algebraic extension of another field

Jakobian

This would show $|T| \leq \sum_{\alpha\in T'} |T_\alpha| \leq |T'|\cdot \aleph_0\leq |T|$

Soumik Mukherjee

Are you talking about lemma $9.26.3$ in stacks project?

Jakobian

yeah, that one

more precisely second part of it

@Jakobian this would then also show that $T'$ needs to be infinite, since otherwise $|T|\leq \sum_{\alpha\in T'} |T_\alpha| < \aleph_0$ so $T$ would be finite

so we'd have $|T| = |T'|$ as desired

$\beta$ is algebraic over $E(T')$ since I can just repeat the argument from earlier that if it were transcendental then $T'\cup \{\beta\}$ would be algebraically independent but $\beta\notin T'$

Now where I'm confused is that they say $E(T')$ is algebraic over $E(\bigcup_{\alpha\in T'} T_\alpha)$ in my notation.

while I know what it means for an element to be algebraic over a subfield of a field (in this case the field is $F$)

I'd need someone to tell me what it means for $E(T')$ to be algebraic over $E(\bigcup ...)$

I suppose it might mean that all $x\in E(T')$ are algebraic over $E(\bigcup ...)$

another issue is that they are using some kind of transitive property that if $\beta$ is algebraic over $E(T')$, and $E(T')$ is algebraic over $E(\bigcup ...)$, then $\beta$ is algebraic over $E(\bigcup ...)$

This seems to me like it's some kind of theorem or lemma used here

because, on my fingers, sure, I know $p(\beta, \alpha_1, ..., \alpha_m) = 0$ for some $\alpha_i\in T'$ and then $q_n(\alpha_i, x_1, ..., x_N) = 0$ for some $x_i\in \bigcup ...$, where $q_n, p$ are non-zero polynomials

but then trying to work explicitly with this, I'd have to show there is some polynomial $P$ such that $P(\beta, x_1, ..., x_N) = 0$

well probably you see why I'm struggling to this with no theory being laid out to me whatsoever

to be clear, what I know: what it means for an element to be transcendental/algebraic over a subfield, what it means to be algebraically independent, transcendence basis

and that's it

Soumik Mukherjee

13

Q: Transitivity of Algebraic Field Extensions

Consider the fields $F, E,$ and $K$, where $F \subseteq E \subseteq K$. If $E$ is algebraic over $F$, and $K$ is algebraic over $E$, show that $K$ must be algebraic over $F$. I know this is a well-known, proved theorem, but I'm trying to understand it on my own. If $E$ is algebraic over $F$...

@Jakobian yes

Jakobian

19:21

@SoumikMukherjee I don't understand this

what is some standard reference for fields?

Roman?

shintuku

@Jakobian Galois Theory by Ian Stewart has full solutions, Contemporary Abstract Algebra by Gallian also has full solutions

Jakobian

wdym by solutions

I'm not going to do exercises from that book

shintuku

all exercises are in the solution manual

Soumik Mukherjee

Suppose $F \subseteq E \subseteq K$, if $\alpha \in K$ is algebraic over $E$ then it satisfies some polynomial with coefficients in $E$. Now if $E$ is an algebraic extension of $F$ then each of the coefficients satisfies some polys with coeff in $F$. So $\alpha$ is algebraic over $F$

shintuku

lang chapter 8 has transcendence bases

Soumik Mukherjee

19:33

@Jakobian You can check this notes math.tifr.res.in/~publ/pamphlets/galoistheory.pdf

@Jakobian Galois theory by David Cox

geocalc33

20:36

Can you isometrically embed $\Bbb H^2 \times (0,1)$ into $\Bbb R^4$? I know that you can isometrically embed $\Bbb H^2$ into $\Bbb R^4$ by the Nash-Kuiper theorem and that you cannot isometrically embed $\Bbb H^3$ into $\Bbb R^4$ by the same theorem

I would say probably not

Jakobian

@geocalc33 by $\mathbb{H}$ do you mean quaternions?

Jakobian

21:26

I think you might mean the half-plane?

shintuku

22:50

@Jakobian tbf I have no clue why they're saying $\beta \in B\B^*$ is algebraic over $F(B')$, but the proof in Ian Stewart's Galois Theory is straightforward

oh nvm, since $B'$ is a maximal algebraically independent subset, any adjunction of a transcendental element must be algebraic over it

follows from lemma 18.4 in Stewart's book

but otherwise using Stack project's own definition, it's just by definition of a transcendence basis

MathCrackExchange

23:44

@shintuku hey

:D

shintuku

hi

MathCrackExchange

How far did you get in AG or Algebra etc?

amazon.com/Theories-Sites-Toposes-mathematical-topos-theoretic/…

I ordered that book a hard copy

It's so cool, compared to Johnstone's ancient books

I have it now :)

shintuku

currently attempting to link dimension and noether normalization, after that will take a break and start differential geometry

MathCrackExchange

It actually teaches you things instead of assuming you're a researcher

@shintuku that's neat. You're way ahead of me

Great job! It will pay off one day

shintuku

i've been told that's the last big step to have enough commutative algebra to comfortably begin AG

MathCrackExchange

23:48

Probably true, otherwise you feel dirty just blindly using the foundational theorems as blackboxes

I'm going to go through Cats & Sheaves - have a torn up hardcopy of that. It also teaches CA

shintuku

@MathCrackExchange sounds cool, topos theory sounds cool

MathCrackExchange

It is very cool. It supposedly has connections to AG, the book is about geometric theories or something like that

One day we will be at a seminar, and we'll be like I know you from MSE!!!

When we're old and gray

shintuku

heheh

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