How can I describe the points of [math]\operatorname{Spec} \mathbb{Z}\left[

File: iwantmom.jpg (90KB, 658x632px) Image search: [Google]

90KB, 658x632px

pleb here, what field of math is this?

>>8124842
Algebraic Geometry

>>8124842
algebraic geometry

its the field where people who dont understand geometry hide

>>8124842
It's a ring, not a field. The spectrum of a field would be trivial.

>>8124838
prime numbers and irreducible polynomials in the variables.

(an ideal is prime iff the quotient by it has no zero divisors)

>>8124838
>That's actually Grothendieck

>>8124871
No, because Z[X] is not a PID (Z is not a field). for instance, every maximal ideal in Z[X] contains a prime integer.

>>8125181
Just to add on for the sake of more information, the original guy is just missing some of the points. In particular, ideals of the form (p,f) where p is a prime integer and f is irreducible mod p. Surprisingly, Spec Z[x] has dimension 2. Adding more indeterminates to your ring amounts to a product of schemes, so this case does tell you a lot.

>>8124838
Don't know. Well as that ring is Jacobson the prime ideals are intersection of maximals, and maximals are generated by a prime in [math] \mathbb{Z}[/math] and some polynomials over the field of p elements (the quotient should be a finite field). I don't know shit about algebraic geometry so I'm sure there's a lot more you can say when you have three variables.

is latex broken or is it just me? also does \operatorname provide some spacing that \mathrm doesn't?

>>8125483
It's broken for me as well, and yes.

>>8124862

not a ring, circlejerk rather

>>8126153

and we have a full autistic spectrum of this circle jerk

come and join the virgin singing!

https://www.youtube.com/watch?v=GJ1S3tFaImo

File: 1453566760447.jpg (55KB, 800x600px) Image search: [Google]

55KB, 800x600px

>>8126196
joining

https://www.youtube.com/watch?v=BipvGD-LCjU

>>8125201
Ok so for an alg. closed field, [math]\operatorname{Spec} k\left[ {{x_1},..,{x_n}} \right][/math] just looks like [math]{k^n}[/math].

Now I know this clearly isn't true for [math]\mathbb{Z}[/math], but how much more complicated would like [math]\operatorname{Spec} \mathbb{Z}\left[ {{x_1},..,{x_n}} \right][/math] compared to the lattice [math]{\mathbb{Z}^n}[/math]? Would they be at all similar?

File: crain.gif (90KB, 480x270px) Image search: [Google]

90KB, 480x270px

>>8126210

oh, look the devil is changing the speed of my typing, the bastard did it again!

I wonder what fedoras think of Alexender > muh feelings?

Are they even more euphoric? So did he go crazy or the Devil exists?

File: spec_z[x].png (86KB, 1007x717px) Image search: [Google]

86KB, 1007x717px

>>8126220
Do you mean geometrically? Well, here's a picture of Spec Z[x] if you haven't seen it before. As you can see, Spec Z[x_1,...,x_n] will be a quite bit more complicated than Z^n. For example, you have points coming from ideals like (x^2+x+1,y^2+y+1,z^2+z+1,2) in Spec Z[x,y,z]. The idea is points correspond to prime numbers, irreducible polynomials over Q, and a prime number together with polynomials irreducible mod p. It's a mess.

>>8124842
Take your pedophile cartoons back to >>>/a/.

>>8126196
this song speaks to me

>>8126249
oy this fucking picture again. never understood it. so prime ideals=generic points and maximal ideals=points.

>>8126612
Nah, (0) is the only generic point. A generic point is one whose closure is the whole space. Basically, every nonempty open set contains (0), so it's close to every single point. Think of it as being everywhere at once.

The non-maximal primes in this example do have some similar properties. For instance, every neighborhood of (2) contains (2,f) for any f irred mod 2 as well. That's why there's that little fuzz drawn around the prime integers. They're everywhere on the line V((2)) in the picture.

>>8126249
essentially was gonna be my answer. my guess is it's a fucking mess. you can find some principle ones through eisenstein's criterion, and you know the usual primes will be in there, but mostly nasty as fuck facts about system of diophantines is gonna be your only hope at organizing it.

>>8124838
prime ideals, son

>>8126651
the question is what do they look like you imbecile

>>8126666
like the prime numbers, except for the ring of polynomials over the integers

also, checked

>>8126689
not prime ideals in general you fucking retarded monkey, the ideals of THIS ring with THIS topology

File: 1460539766386.png (114KB, 816x679px) Image search: [Google]

114KB, 816x679px

>>8126693
now thats not a very good attitude anon

File: 4733010-8380799852-dukex.jpg (59KB, 490x519px) Image search: [Google]

59KB, 490x519px

>>8126693

to be a good scientist you must be like pic related

>>8126666
total waste of tetrahex

>>8124838
They look like the points of an affine scheme.