I never did the symmetric polynomial approach, Artin's is much easier. I don't know that it would be true that every quantic polynomial's galois group would be S_5

quintic*

For $p,q,r,s$ in some field, how would I prove from the field axioms that $\dfrac{p}{q}\cdot\dfrac{r}{s}=\dfrac{pr}{qs}$, i.e., $(pq^{-1})(rs^{-1})=(pr)(qs)^{-1}$?

every irreducible one perhaps

($q,s\neq0$ here)

Is there something I can do to make latex commands show up in these chat rooms. i keep getting the source language instead of the rendered output

@DavidReed Look up "ChatJax"

ty

no, look at the link to LaTeX in chat in the upper right @DavidReed

@TedShifrin The top search result (for me at least) gives the same link, so ¯\_(ツ)_/¯

oh, I guess it depends where you search, Legion

But yeah, would you have any idea how I'd solve my current problem?

The equation seems pretty basic

@LegionMammal978 first show that $b^{-1}a^{-1}$ is an inverse of $ab$ (this works in any group)

the rest follows from commutativity and associativity

I guess you also need to use the uniqueness of inverses

but I think if you use the notation $a^{-1}$ you already asumme that

This seems sort of silly to me. Multiplication is commutative and associative ... so ... done.

is this the regex for this language: x starts with 0 and does not contain substring 101

I'm just not sure how to get from, as @Mathein said, from $aa^{-1}=1$ to $a^{-1}b^{-1}=(ab)^{-1}$

0*(1*000*)*1*0

@LegionMammal978 what happens if you multiply $ab$ and $a^{-1}b^{-1}$?

Well, what is $(ab)(a^{-1}b^{-1}) = a(a^{-1}b^{-1})b$?

Assuming you've defined a field to be a commutative division ring via the notion of group of units, uniqueness is implied. mathein's approach is the correct way.

Ah, I see now

Lots of commutativity and associativity

That's what I said :P

Granted, I was a bit short.

@Theo The string 01 is contained in the language but does not match your regex

@LegionMammal978 why not? since (1*000) does not have to be present, it can show up zero times

there we are left with 0*1*0

@Theo It does not contain the final 0

@DavidReed the notion of a "general polynomial" is a bit strange. The general polynomial over a field always has Galois group $S_n$ even if there is not polynomial over the field with Galois group $S_n$

I just proved that the set of all zero divisors, along with $0$, in a commutative ring forms an ideal. Is this right, or did I make a mistake?

this is wrong

consider $\Bbb Z/(6)$

$3$ and $2$ are zero-divisors

products work, but not sums :)

but $3-2$ is not

Ah...I see...

@MatheinBoulomenos Oh ok. I don't believe I've ever encountered that terminology. Thank you for letting me know.

@LegionMammal978 what about this? 0*(1*000*)*1*0*

@Theo That matches the empty string, which is not in the language

Okay...If $d$ is a zero divisor, then will the principal ideal $(d)$ consist entirely of zero divisors?

yes

@LegionMammal978 0*(1*000*)*1

that should do it i believe

@user193319 Good other rules to know for ideals: $(a) = (b)$ iff $a,b$ are associates and $(d) = R$ iff $d$ is a unit

@Theo Doesn't match 0.

the implication $(a) = (b)$ implies $a$ and $b$ are associates only works in integral domains iirc

iirc?

if I recall correctly

ah

You may be right, when I went through algebra the notion of an ideal was only defined for elements of integral domains

If that's true then I stand corrected.

@LegionMammal978 this? 0*(1*000*)*0

@LegionMammal978 nvm doesn't cover 01

@LegionMammal978 is this possible? 0(0*)(1*000*)*

@Theo Doesn't match 01.

why not? 0(no 0 after)(one 1, no 000)

That would be the regex 00*(1*(000)*)*

And that doesn't match 010

is it possible to do: 0(1*0*)*

that way 101 won't occur, and 01, or 0000, or 010 still occurs

@Theo That matches 0101.

oh right

okay im a bit lost, I know 0*(1*000*)*1*0* will make it so 101 does not occur in substring

but how do i avoid an empty string

and make sure it starts with 0

gimme a minute

hey

anyone like graph theroy?

anyone still here? :-o

I haven't gone through graph theory in nearly a decade

My guess is if nobody here is up for it, comp sci ppl use it a lot, you may have luck in stack overflow chat