Atif Degirmendzic https://d-atif.github.io/output/index/ index /output/index/ Subtreehouse YooWelcome to da forest.Who: Atif DegirmendzicAll the stuff Atif Degirmendzic2026615https://d-atif.github.io/output/0002/0002/output/0002/List of StuffAtif Degirmendzic202673https://d-atif.github.io/output/0006/0006/output/0006/Intro to Galois theory, part 1: Algebraic (Field) ExtensionsNote let me try and summarize the section Algebraic Extensions from Malik's fundamentals of abstract algebra. ill first outline the big picutre: what role this section has in galois theory, then ill focus on the small picture: how do we build (algebraic) extensions of fields. so we are still at the low stages of the theory building that will culminate in galois theory. i find it hard to even start talking about the point of this section without spoiling the point of galois theory, thus mild spoiler alert: galois theory gives us something useful about certain posets that we care about. (actually it's not posets but lattices, but close enough). what i will do here is talk about these "posets", the "something useful" part i wont be going into here. we'll be working with fields. instead of saying that " is a subfield of ", we'll be saying that " is field extension of ". (writing just saves us a lot of time, but there are other benefits). we also read as " over ". example of usage: "let be a field and it's extension" = "let be a field and a field s.t. is a subfield of ". the reason this notation makes this reversal is because we will indeed be starting with and constructing the field from ; so we ascend from to .(spoiler: will usually be a certain field that we denote with , which does a better job at hinting that the extension is something that we actually built on top of ) id say the main result of this section was the following: it turns out this "over" relation between and is a partial order, even though at this point we don't explicitly frame it that way. same with different words: the "extension" relation induces a poset on . this is the poset that galois theory works its magic on. (i think this is roughly correct, im writing this for myself mostly so sorry not sorry for misleading stuff.) so what's the deal with this poset? well we care about a specific kind of extension, namely a so called algebraic extension. ok we know means is an extension of . we say that is an *algebraic* extension of K if every element in is algebraic in . say is algebraic in , that means the following: there's a polynomial s.t. . that's it. a is algebraic iff it's a zero of a polynomial with coefficients in . so i might've lied earlier (unsure!), it's actually the relation "algebraic extension" that induces our poset, not merely the "over"/"extension" relation. one of the very last theorems we prove is basically transitivity for this relation: Theorem: is algebraic iff and are algebraic. (i omit the assumption that and are extensions of , and is an extension of , thus i say just "algebraic".) what we said so far was contextualization for the big picture. so let me recenter this summary at this point and focus on the small picture of this section. this section is about algebraic elements and about algebraic extensions of fields. say i give you the field , and say you stumble upon the innocent looking polynomial . you notice that , but . weird huh. may i suggest extending the field via , that we just saw is an algebraic element? but how do we do that? so this section is about how we construct extensions of fields and algebraic extensions in particular. even more particular: it's about *extending* a given field via an algebraic element , so we want a field that contains .(remember that is not in , was not in ; all elements of are algebraic in (why? of what polynomials in are they solutions of?)). this is now the usual situation where doesn't preserve the algebraic structure, ie isn't a field (trust). but using the familiar trick/notation for "generated by", is a field that that contains . so let's unpack what actually is. it's the following thing: . and as one might remember, is the same as the intersection of all fields (or any algebraic structure) that contain the field , formally: . connecting this with what we just said we have the following . the meaning of this is that is the *smallest* algebraic extension of via . and we are glad that it's smallest because any surplus would be expensive, we want to pay only for what we have to pay for; we want to be efficient (see: adjoints wiki, adjoint.tree (coming soon)). this also sheds a very bright ray of light on the purpose of polynomial rings, which i shall say more about later. Atif Degirmendzic2026626https://d-atif.github.io/output/0005/0005/output/0005/Point free group theory?Thoughtare there approaches to group theory that are a bit more global in the following sense: in the category set objects are sets. there are no constraints on what a subset (ill call it the substructure) of a set can be, as long as it satisfies the basic definition (contains only some (or all) elements of the parent set). in the category grp, the objects are groups, and we know that the order of the subgroup (subgroups are now the relevant substructure) has to divide the order of the group; however we know this is not sufficient, since no real analogue of lagrange theorem exists. so im wondering if group theory can be built by putting a constraint on what its substructure (subgroup) can be ? are there approaches to group theory like this that are similar in spirit?what kind of category would we get if we just said: cardinality (order) of substructure has to divide the cardinality (order) of parent structure? i assume the category of finite groups would be a "subcategory" of this? has anyone done this? (see: fusion categories)i really dont know much about it, but this is reminiscent to me of point free topology. is there point free group theory ?(ofc the natural question onward would be: is there point free [insert structure here] theory?)note: this came to mind after a brief convo w Asija, i was trying to explain why being able to see a group G as a subgroup of group of permutations is useful (subgroups divide order etc.) Atif Degirmendzic2026524https://d-atif.github.io/output/0004/0004/output/0004/Bethe lattice and tesselations/fundamental polygonsThoughtis there a manifold that corresponds to a eg regular tree tesellation of the hyperbolic plane? if theres no direct correspondence, is there a way to construct a manifold thats in some way related to the tesselation? (myb take the complement, rotate, glue, remove/add sides in a meaningful way, and so on) are there discrete manifolds? is there perhaps a correspondence between discrete manifolds and the manifolds that we can (in some sensible way) associate with a tree that tesselates the hyperbolic plane can we tesselate hyperbolic plane with ? probably not a tesselation, but. a spiral? 🌀 (tangential stuff below) do we get anything if we try to define a curve as a graph with uncountably many points? or thats how we start, then we impose additional structure on it, that we need ie top and geo structure. question: would the graph aspect perhaps eliminate some pathologies? would it eliminate interesting curves? can we solve wave equations on those (discrete) manifolds? would it have any connections with the solution of the graph? there is probably patternally similar stuff out there, ie, do something for the object that tesselates (here: our tree; elsewhere: a square, or octagon), you get ~info about the manifold (here:?, elsewhere: torus, 2-torus); euler characteristic is one such thing i think. can we solve wave equations on a torus or 2-torus? are the wave and heat equations part of some well known class of equations? bethe lattices (“occur”) can be viewed as discrete subgroups of fuchsian groups (theyre also lattices in the sense of a lattice in a Lie group). modular group is a fuchsian group. when the modular group acts on , it tesselates , the orbifold that corresponds to that tesselation was drawn by Lejla… two cusps, infinite neck thing. lets pretend that some bethe lattice corresponds to the modular (fuschsian) group (in worst case scenario, i hope we can find for a specific q a group that yields orbifolds as the modular group does). now, perhaps the corresponding bethe lattice subgroup of the modular group induces a coarser tesselation of the tesselation by the whole modular group. what’s the corresponding orbifold/manifold of that tesselation? let’s solve wave equation on that! and investigate if theres a connection with the kernels of the wave eq on the regular tree problem s ovom pricom je sto (valjda?) obicno korsitimo fundamnetalni poligon da dobijemo info o manifoldu, jer je poligon jednostavniji objekat. ili ipak ide u oba pravca? Atif Degirmendzic2023211https://d-atif.github.io/output/0003/0003/output/0003/Where do ODEs come from?LinkSee this: https://youtu.be/B0E0q6ICK3c?si=Q-l8l-j4ORr9qRN9&t=957 References Context Backlinks Related Atif Degirmendzic 2026 6 15 https://d-atif.github.io/output/0001/ 0001 /output/0001/ All the Stuff Atif Degirmendzic 2026 7 3 https://d-atif.github.io/output/0006/ 0006 /output/0006/ Intro to Galois theory, part 1: Algebraic (Field) Extensions Note let me try and summarize the section Algebraic Extensions from Malik's fundamentals of abstract algebra. ill first outline the big picutre: what role this section has in galois theory, then ill focus on the small picture: how do we build (algebraic) extensions of fields. so we are still at the low stages of the theory building that will culminate in galois theory. i find it hard to even start talking about the point of this section without spoiling the point of galois theory, thus mild spoiler alert: galois theory gives us something useful about certain posets that we care about. (actually it's not posets but lattices, but close enough). what i will do here is talk about these "posets", the "something useful" part i wont be going into here. we'll be working with fields. instead of saying that " is a subfield of ", we'll be saying that " is field extension of ". (writing just saves us a lot of time, but there are other benefits). we also read as " over ". example of usage: "let be a field and it's extension" = "let be a field and a field s.t. is a subfield of ". the reason this notation makes this reversal is because we will indeed be starting with and constructing the field from ; so we ascend from to .(spoiler: will usually be a certain field that we denote with , which does a better job at hinting that the extension is something that we actually built on top of ) id say the main result of this section was the following: it turns out this "over" relation between and is a partial order, even though at this point we don't explicitly frame it that way. same with different words: the "extension" relation induces a poset on . this is the poset that galois theory works its magic on. (i think this is roughly correct, im writing this for myself mostly so sorry not sorry for misleading stuff.) so what's the deal with this poset? well we care about a specific kind of extension, namely a so called algebraic extension. ok we know means is an extension of . we say that is an *algebraic* extension of K if every element in is algebraic in . say is algebraic in , that means the following: there's a polynomial s.t. . that's it. a is algebraic iff it's a zero of a polynomial with coefficients in . so i might've lied earlier (unsure!), it's actually the relation "algebraic extension" that induces our poset, not merely the "over"/"extension" relation. one of the very last theorems we prove is basically transitivity for this relation: Theorem: is algebraic iff and are algebraic. (i omit the assumption that and are extensions of , and is an extension of , thus i say just "algebraic".) what we said so far was contextualization for the big picture. so let me recenter this summary at this point and focus on the small picture of this section. this section is about algebraic elements and about algebraic extensions of fields. say i give you the field , and say you stumble upon the innocent looking polynomial . you notice that , but . weird huh. may i suggest extending the field via , that we just saw is an algebraic element? but how do we do that? so this section is about how we construct extensions of fields and algebraic extensions in particular. even more particular: it's about *extending* a given field via an algebraic element , so we want a field that contains .(remember that is not in , was not in ; all elements of are algebraic in (why? of what polynomials in are they solutions of?)). this is now the usual situation where doesn't preserve the algebraic structure, ie isn't a field (trust). but using the familiar trick/notation for "generated by", is a field that that contains . so let's unpack what actually is. it's the following thing: . and as one might remember, is the same as the intersection of all fields (or any algebraic structure) that contain the field , formally: . connecting this with what we just said we have the following . the meaning of this is that is the *smallest* algebraic extension of via . and we are glad that it's smallest because any surplus would be expensive, we want to pay only for what we have to pay for; we want to be efficient (see: adjoints wiki, adjoint.tree (coming soon)). this also sheds a very bright ray of light on the purpose of polynomial rings, which i shall say more about later. Atif Degirmendzic 2026 6 26 https://d-atif.github.io/output/0005/ 0005 /output/0005/ Point free group theory? Thought are there approaches to group theory that are a bit more global in the following sense: in the category set objects are sets. there are no constraints on what a subset (ill call it the substructure) of a set can be, as long as it satisfies the basic definition (contains only some (or all) elements of the parent set). in the category grp, the objects are groups, and we know that the order of the subgroup (subgroups are now the relevant substructure) has to divide the order of the group; however we know this is not sufficient, since no real analogue of lagrange theorem exists. so im wondering if group theory can be built by putting a constraint on what its substructure (subgroup) can be ? are there approaches to group theory like this that are similar in spirit? what kind of category would we get if we just said: cardinality (order) of substructure has to divide the cardinality (order) of parent structure? i assume the category of finite groups would be a "subcategory" of this? has anyone done this? (see: fusion categories) i really dont know much about it, but this is reminiscent to me of point free topology. is there point free group theory ? (ofc the natural question onward would be: is there point free [insert structure here] theory?) note: this came to mind after a brief convo w Asija, i was trying to explain why being able to see a group G as a subgroup of group of permutations is useful (subgroups divide order etc.) Atif Degirmendzic 2026 5 24 https://d-atif.github.io/output/0004/ 0004 /output/0004/ Bethe lattice and tesselations/fundamental polygons Thought is there a manifold that corresponds to a eg regular tree tesellation of the hyperbolic plane? if theres no direct correspondence, is there a way to construct a manifold thats in some way related to the tesselation? (myb take the complement, rotate, glue, remove/add sides in a meaningful way, and so on) are there discrete manifolds? is there perhaps a correspondence between discrete manifolds and the manifolds that we can (in some sensible way) associate with a tree that tesselates the hyperbolic plane can we tesselate hyperbolic plane with ? probably not a tesselation, but. a spiral? 🌀 (tangential stuff below) do we get anything if we try to define a curve as a graph with uncountably many points? or thats how we start, then we impose additional structure on it, that we need ie top and geo structure. question: would the graph aspect perhaps eliminate some pathologies? would it eliminate interesting curves? can we solve wave equations on those (discrete) manifolds? would it have any connections with the solution of the graph? there is probably patternally similar stuff out there, ie, do something for the object that tesselates (here: our tree; elsewhere: a square, or octagon), you get ~info about the manifold (here:?, elsewhere: torus, 2-torus); euler characteristic is one such thing i think. can we solve wave equations on a torus or 2-torus? are the wave and heat equations part of some well known class of equations? bethe lattices (“occur”) can be viewed as discrete subgroups of fuchsian groups (theyre also lattices in the sense of a lattice in a Lie group). modular group is a fuchsian group. when the modular group acts on , it tesselates , the orbifold that corresponds to that tesselation was drawn by Lejla… two cusps, infinite neck thing. lets pretend that some bethe lattice corresponds to the modular (fuschsian) group (in worst case scenario, i hope we can find for a specific q a group that yields orbifolds as the modular group does). now, perhaps the corresponding bethe lattice subgroup of the modular group induces a coarser tesselation of the tesselation by the whole modular group. what’s the corresponding orbifold/manifold of that tesselation? let’s solve wave equation on that! and investigate if theres a connection with the kernels of the wave eq on the regular tree problem s ovom pricom je sto (valjda?) obicno korsitimo fundamnetalni poligon da dobijemo info o manifoldu, jer je poligon jednostavniji objekat. ili ipak ide u oba pravca? Atif Degirmendzic 2023 2 11 https://d-atif.github.io/output/0003/ 0003 /output/0003/ Where do ODEs come from? Link See this: https://youtu.be/B0E0q6ICK3c?si=Q-l8l-j4ORr9qRN9&t=957 https://d-atif.github.io/output/atifdegirmendzic/ atifdegirmendzic /output/atifdegirmendzic/ Atif Degirmendzic Person https://d-atif.github.io PMF Student Contributions