which is better groups or fields?

>>7972314
better in terms of what?

>>7972314
test for name

>>7972314
Finite fields > Groups >>>> All other fields

What the fuck do you even prove with groups or fields? I don't get it.

Rings.

FIELDS ARE GROUPS.

Anyway an understanding of both concepts is required for Galois theory.

>>7972323

Where math is concerned, "better" usually means more useful/can be applied to more and different areas, etc. So to answer OP's question I'd say groups because the idea is more general and applies to structures more often. Fields are important too though, otherwise we wouldn't have abstracted them.

>>7972360

Seemingly disparate algebraic structures, or things that give rise to algebraic structures, turn out in the end to be the same, and can be proven to be such. We therefore no longer distinguish between the "two" cosmetically different (i.e. not actually different at all) algebras. Groups/rings/fields are among the simplest examples of algebraic structures, and once they've been learned, it becomes possible to compare two different ones, per the above.

>>7972382
>fields are groups
Yeah, because there's definitely an isomorphism between the two categories
/sarcasm

>>7972421
Holy fuck you are retarded.

>>7972314
Groups, a Field isn't even an algebra over a monad.

>>7972445
Listen. I know you were trying to allude to the Galois correspondence. But listen: it is you who are retarded. It is not even known if every finite group is the Galois group of a field extension.

So please, illuminate me on your correspondence between fields and groups.

>Holy fuck are you retarded

>>7972531
A field is an abelian group with another binary operation. Were you seriously unable to interpret the statement correctly?

>>7972531
you mean it is not known over [math]\mathbb{Q}[/math]

>>7972552
His post is missing the point though, and the category statement was a pretty good comment.

Rings

>>7972552
So fields are groups in the same way groups are their underlying sets.

This is some Poe's Law level of retardation. Can't tell if satire.

>>7972552
As >>7972561, I realize now what you are trying to say, and what you are trying to say arises from a fundamental misunderstanding of the formalisms. Yes, the underlying set of a field can be endowed with the group structure induced by the field's additive binary operation, but by no means is a field a group. Model-theoretically, groups and rings have totally different signatures — they could scarcely be more different types of objects. Never say "fields are groups" or "fields are a type of group" unless you want to reveal your rather egregious fundamental misunderstandings as to the nature of these mathematical objects.

>>7972552
And, as the prior two posters, let me add: Don't call others retarded on scientific/mathematical matters unless, at a minimum, you truly know what you're talking about. Otherwise you embarrass yourself.

>>7972577
Might want to change wiki then
Fields are a ring whose nonzero elements form a commutative group wrt multiplication
Rings are a commutative group with blah blah

>>7972531
>talks shit about others
>confuses trivial problem with inverse galois problem
fäm…

>>7972580
The first one is accurate.

Show me where on wiki it says rings are groups.

One can say "A ring is a group endowed with additional structure", but this endowment makes it no longer a group.

>>7972587
I had difficulty wrapping my mind around his stupidity. How the hell is one supposed to interpret "fields ARE groups"? I had to really step back into the neophyte mindset.

>>7972561
>>7972577
>>7972578
Kek I'm still correct with my statement. Ask any of your professors if a field is a group and they'll confirm. Yes, there are differences that arise from the added restrictions on fields. Similarly, functions are sets of ordered pairs, yet a function is very different from some arbitrary set of ordered pairs. That does not make the statement any less informed. This is called abstraction. It shows are somewhat different objects can behave similarly under certain operations.

Beyond that, my first post was >>7972552 so you don't need to get your dick tied in a knot.

Fields definitely

>>7972584
Ahh right because type has a formal definition in abstract algebra?

>>7972594
A field is NOT a group.

Ask any of your professors if a field is a group.

I mean, as >>7972599, seriously.

Seriously.

I can't wrap my head around this retardation.

"Fields are a type of group."

Christ.

>>7972421
This. What a retard.

>>7972593
>>7972593
From wiki
>A ring is an abelian group with a second binary operation that is associative, is distributive over the abelian group operation, and has an identity element.

> One can say "A ring is a group endowed with additional structure", but this endowment makes it no longer a group.

Whats the difference?
Also was I wrong in my proof when I wrote:
the set of sequences that converge to zero is a subgroup of the ring of Cauchy sequences.

>>7972594
My profs would laugh at your retard ass for not knowing what a formal definition of an algebra is.

>>7972612
Because you retard. You can consider a group as an algebra over a monad but you can't do the same with a field because fields are shit.

GROUP > field

>>7972612
I can understand how that phrasing could confuse you.

Read it as:
>A ring is an abelien-group-with-so-and-so-additional-structure
Thus the object of the sentence is the entire clause, rather than just the word "group".

And the entire object clause does not describe a group, but some other type of object — a "group-endowed-with-additional-structure", which we call a ring.

>>7972615
Dont have to be a jerk about it.

>>7972594
Nope. They are different types of objects.
Groups are an ordered pair (Set, Binary operation)
Fields are (Set, Binary Operation, Other Binary Operation)
Again, defined as different objects. Yes,every field 'has' a group in it, but it's basically an uninteresting a abelian group.

The answer is obviously groups.

Groups are the more general concept and although fields are are often quite important for most branches of mathematics groups are just so much more versatile.

There are fundamental groups, cohomology groups, stabilizer groups and even groups to describe field extensions aka galois groups.

The only fields who get much attention are the reals, compley, rational and possibly some p-adic closures of the rationals.

In group theory there are more types of groups than that, we have cyclic, abelian, nilpotent, solvable, simple and the list goes on and on and they all have examples who appear in a useful context one way or another.

Get fucked fieldfags.

>>7972642
>The only fields who get much attention are the reals, compley, rational and possibly some p-adic closures of the rationals.
Ever heard of GLn? How oblivious are you? Don't listen to this guy.

>>7972652
Fuck off, nobody cares about F_p. I mean look at it, it's not even infinite, neither is it complete or algebraically closed. You can't do topology with it, you can't do analysis with it.

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>>7972642
>he's never heard of a single field between the rationals and the reals
>he doesn't realize that studying these fields led to the notion of a group

>>7972675
>what are symmetry groups
>what are permutation groups

Seriously though, why do you plebs finally step it up and just work with magmas. All those extra axioms just hold you back.

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>>7972314
Lattices

>>7972675
This is damn good bait, I gotta admit.

>he's never heard of a single field between the rationals and the reals
lol

>he doesn't realize that studying these fields led to the notion of a group
Dear Tyrone Mann, people had the notion of a group before people had a notion of mathematics.

>>7972682
>what are symmetry groups
>what are permutation groups
Yes! Great examples of groups discovered through the study of field extensions

>>7972686
That's not how it happened d00d.

>>7972682
Why not work with objects that have no structure at all? Structure just holds you back.

>>7972642
Do you know about function fields?

>>7972692
No structure is nothing. A single binary operation is no more and no less than you need to do anything of import.

>>7972733
All you really need is a single binary relation and you can get whole universes of set theory.

All of mathematics can be simulated in a magma satisfying translations of the axioms of set theory.

>>7972622
>Yes,every field 'has' a group in it, but it's basically an uninteresting a abelian group.
So much this. Fields are boring plebby things. Groups are where it's at.

>>7973667
The abelian group structure is not what makes fields interesting, though.

>>7973885
>not even the abelian group structure manages to make fields interesting though.
Fixed

Idempotent semirings >>>>> all.
Prove me wrong.

>>7974550
Don't cut yourself on those edges.

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>>7974550
strict total order > all

What about frobenioids?

A Frobenioid is a category with some extra structure that generalizes the theory of line bundles on models of finite extensions of global fields.

>>7972399
This.

>>7972605
Lol. Everybody above this post is retarded.

Fields are a type of of group.

It was so fucking obvious but completely misunderstood.

>>7974665
>what's better, a group or a field?

HEY GUYS WHAT ABOUT FROBENIOIDS WHY YOU NOT CONSIDER FROBENIOIDSSS!!

>>7974677
You are a fucking retard. If a field is a group then describe the monad that gives it as an algebra. Oh wait, there isn't one because fields are shit.

>>7975363
>I use complex words so that people will think that I am not an undergraduate

>>7975403
Monads aren't that hard desu. But to use them in this particular argument is unbelievably silly, since the whole thing just comes down to semantics.

>>7975425
*although the guy who says fields are a type of group is wrong because he uses the word 'type'

>>7975425
No way man, not silly when someone is trying to claim fields are groups.
http://mathoverflow.net/questions/3003/in-what-sense-are-fields-an-algebraic-theory

>>7972674
>Fuck off, nobody cares about F_p

You can algebraically close any finite field, and Robinson's theorem shows that the resulting fields do have similar properties to C. Working in ACFp has interesting number-theoretic consequences.

Topology has other uses besides traditional analysis. Setting ACF_p=k and looking at k-algebras is a perfectly good setting for algebraic geometry (so things like k[x] have a natural topology on them). You can then define "algebraic tangent spaces" and so forth, if you want to look at analogues of "analytic" results.

Lattice theory is best theory

>>7976158
agrizzled

>>7972593
>I had to really step back into the neophyte mindset.
Nah, you're just genuinely retarded.

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>yfw you first learned that all finite fields of a given order are isomorphic

>>7976489
That's just a side effect of how uninteresting they are.

>>7976158
Lattices are tight shit, anon.