So you determine something is a group, field, ring, etc. So what?

>>9003862
>What does that gain you?
The ability to apply theorems about that algebraic structure

>>9003868
Right.

So?

>>9003875
So there's useful theorems and techniques you can use from the theory of the algebraic structure that might help you in solving whatever your problem is or studying this object you've determined fits that structure

What kind of answer are you looking for exactly?

>>9003885
Obviously, I'm not a super math nerd, but with geometry, if you take away Euclid's 5th postulate, you lose things like the sum of angles of a triangle is 180, the pythagorean theorem, etc.

Those are nice tools to do other stuff with.

If I know a ring is a ring, what problems can I solve using theorems about rings that I couldn't before? Does this shortcut any problems?

Does knowing the algebraic property of matrices let me solve more matrix work quicker?

>>9003862
In algebra the answer to any question with "what is this structure good for" is always classification.

>>9003893
For example, modular forms are just complex functions that satisfy certain properties (https://en.wikipedia.org/wiki/Modular_form)

After finding one of them, you might wonder how many others there are. So you look and see out that the set of modular forms of a given weight form a vector space, so you can calculate the dimension of this vector space which tells you how many modular forms there are of that weight. Then you can consider things like maps between vector spaces of different weight and use whatever tools from linear algebra you want

>>9003909
in addition this was a key step in the proof of fermat's last theorem since the construction of an elliptic curve using a fermat triple is supposed to yield a certain (non-trivial) modular form, but then you look at the dimension of the vector space this modular form is supposed to be in and it's 0, which is where the contradiction arises

>>9003893
consider trying to optimize a system of equations with n ode shits. understanding how they behave in general lets you start to figure out what you can do to them and retain your encodings. I think this is called polytopic theory or something. im still learning this stuff myself

>>9003862
Everything is fields. No clue why you think Ring and Group exist.

>>9003893
since you mention the 5th postulate, let me comment on this. we have a natural model for the Euclidean geometry, namely R^2. we have vector operations, the inner product etc. all of which make this into a really comfy object to work with. those three guys then proved that if you remove the 5th postulate, you obtain different kinds of weird geometries, but what exactly are they ? how do I describe them and how do I work with them ? turns out that the essential part of the Euclidean plane is not the set itself, but it is the group of its symmetries, namely the group of transformations preserving angles and distances. all the characteristic things like the sum of angles in triangles are precisely the properties which are preserved by the action of this group. different kinds of geometries are then obtained by replacing this group by a different one. for example the group of transformations preserving only angles leads to the similarity relation for triangles, but not congruence. the fancier stuff like hyperbolic and spherical geometry is more complicated, but it's the same idea: the definition of "a geometry" is that it's some space together with a group which represents the allowed symmetries. furthemore the space itself is basically obtained from the symmetry group by taking a quotient by a certain subgroup. so it boils down to: geometry = a pair consisting of a group and a subgroup. (of course it must a special kind of group, keyword is lie groups). the rabbit hole goes deeper than you think.

>>9003862
Anything you can prove in terms of groups, fields or rings are very general results for they apply do anything that are groups, fields or rings. Instead of proving the same thing for various instances of groups you only have to prove it in terms of Group Theory [math]\textit{once}[/math].

If you're looking for more practical stuff then Burnsides Lemma or Sylows theorem might interest you. Perhaps Fourier Analysis on Groups by Rudin could show you a thing or two.

>>9003893
For the specific example of rings, you can look at the solution to the problem of finding the integers that can be written as a sum of two squares using the Gaussian integers and compare it to a purely computational proof.
Then turn to the proof of the fact that every integer is a sum of four squares, first by elementary means, then using Hurwitz quaternions.

Other examples of situations where structure helps clean up solutions include linear ODEs with constant coefficients and linear recurrent sequences with constant coefficients (it's basically the same thing), linear ODEs with variable coefficients (here, you need some hard analysis but linear algebra still helps), all sorts of counting problems that you can solve by noticing that you are actually trying to find orbits for a certain group action, the classification of conics and quadrics (which becomes much easier once you realize that you are classifying orbits under the action of GL(3) by congruence, which is a purely linear algebraic problem), etc.

>>9003893
When you do geometry, why do you bother classifying say triangles as right, obtuse, or acute? Why do you bother making a distinction between polygons and non-stop polygons? Why do you consider hyperbolic geometry separate from euclidean geometry?

All of these objects have general properties that differentiate them, so it's useful to characterize them. Likewise, algebraic structures have properties in common that make it useful to distinguish between them.

For example, matrices over a field have properties similar to but not quite the same as the real numbers. They can be added and multiplied, but in what ways are these analagous to real number addition and multiplication? Can matrix equations be solved in the same way as equations of real numbers? What about diophantine equations and modular number equations? Ring theory answers all of these questions at the same time.

>>9003862
>Is this just for proving things like Fermat's last theorem and other issues with diophantie equations or …?
That's why group theory was initially formulated for, yes.