#Inverse Galois Problem

[tilde-he]

“finite“ field field extensions with characteristic a nonstandard prime, and the inverse Galois problem

So, the inverse Galois problem: Given a (finite) group, is it the Galois group of some field extension? Or, rather, is there, for every finite group, some field extension which has that as its Galois group?

To quote Wikipedia : “ This problem, first posed in the early 19th century,is unsolved. “

Here is some reasoning that I think makes sense. It does not solve the problem (duh. This is a blog post, not a paper.) but I think it is interesting in relation to the problem.

If G is some group, and if it cannot be proven in (1st order?) Peano arithmetic that G is not the Galois group of any field extension of finite fields, or just of fields of nonzero characteristic (assuming that this statement can be expressed in Peano arithmetic in an appropriate way), then, by the model existence theorem, there is a model of arithmetic in which there is such a field extension.

Now, I don’t exactly “know” model theory, so this next thing I say could be quite wrong, but my impression is that we can have a nonstandard model of set theory where the set whose existence is guaranteed by the axiom of infinity (i.e. our set of natural numbers), is actually the set of elements for the nonstandard model of peano arithmetic that we took.

And, this nonstandard model of set theory is living inside our normal set theory, and its sets are like, also actual sets, and the “is an element of” is the actual “is an element of” of our actual set theory. I think this is called an inner model (however, I’m not sure. Wikipedia seems to say that for it to be an inner model, it has to have all of its ordinals in common? which doesn’t seem to be what is the case in what I’m describing, so maybe it doesn’t count as an inner model.).

Anyway, so, we have our nonstandard model of set theory and of the natural numbers, and in this nonstandard model we have that there is a nonstandard prime p such that there is a field F of characteristic p, and a field extending it, K, such that the field extension K/F is Galois.

Now, if we look at one of these fields in this nonstandard model of set theory, from the perspective of our actual set theory, while it won’t be a field of finite characteristic, it should still be a field, just of characteristic 0 instead. Because, the function for the multiplication, addition, negation, etc. will still all be valid functions from the perspective of the actual set theory, so the field axioms should still be satisfied. (characteristic 0 because there is no natural number n such that 1 + 1 + 1 + ... n times, = 0 ).

And, furthermore, the field extension should still be a field extension, and each of the field automorphisms from before should still be field automorphisms.

However, I think that, when looking at the field extension this way, there may be intermediate fields that aren’t there in the nonstandard model, because, essentially, the power sets don’t actually have to have all the subsets, only the ones that can be like, specified, or whatever.

Similarly, there may be additional automorphisms that don’t exist in the nonstandard model, but do “in reality”.

And, perhaps in our actual set theory, K/F isn’t even a Galois extension!

But, this is no issue!

The automorphisms that we had, that inside the nonstandard model comprised Gal(K/F) , all still work as automorphisms, in our actual set theory, and still form a group, just not necessarily Gal(K/F).

So, if we just take K/(the fixed field of that group) , we will then get a Galois extension, which has as its Galois group, that same group, which is isomorphic to G.

So, if my understanding of model theory isn’t too broken, then: If it can’t be proven that there is no prime p such that G is (isomorphic to) the Galois group of some field extension with characteristic p, then there is a Galois field extension of characteristic 0, the Galois group of which is (isomorphic to) G .

So, the possible “answers” to the inverse Galois problem for a finite group G are: 1) “Yes, there is such a field extension” 2) “No, it can be proven that no field extension produces that group as its Galois group.” 3) “No, and it can be proven that no field extension of positive characteristic produces such a Galois group, but while there is also no such one for characteristic 0, this cannot be proven.”

Actually, come to think of it, if it couldn’t be proven about the characteristic 0 case, then wouldn’t there also be a nonstandard model of, something, where there was one with char 0? Could one then extract that to get an actual one of char 0 as well? I suspect the answer is yet.

So, in that case, I think that this seems to kinda point at a way to show that: If a finite group G is not the Galois group of any field extension, then this fact can be proven about G.

So, that’s nice.

There’s, ... probably a way to prove that without resorting to talking about nonstandard models. This is probably way overkill to show something that people presumably already knew. (Or just wrong, but it seems right to me.)

[jonathankatwhatever]

Watching the dreck that is Babylon. These movies all look and sound like they came from the same factory of excess posing as creativity and obviously staged sequences staged to mimic not being staged, but without the irony of that phrasing.

I know my dream has been to calculate using Hexes and Triangles, so that connects to grid squares. I feel like I’m getting closer. I could say the first part of that sentence easily, but there was a hang up at the word ‘connects’ because I have no idea how whirling dials connects. I suppose we can try to run through it. I guess we should start with CM100, because we can build it easily and it has certain properties which seem really important.

And that means what? Aside, it’s interesting watching Margot Robbie because her gift is that she connects to the observer rather than to the other actors, so it’s extremely natural and extremely staged, sort of that old-time energy like Hepburn had doing the same sort of thing.

I would love to see you as a mastermind. You would be very believable.

I keep thinking that certain problems can’t be solved and that should be obvious because we should be able to identify basic dimensions, which means once you get to 5, then you’re done with general solutions.

In fact, you can see why 4 is hard by looking at Winding. A clutch of anxiety: do we believe we know this? Do we believe when believing is how we’ve found every answer? So 4 is hard because it’s possible to generate a solution because we’re only 1 step from SBE, meaning we can still count SBE, can still count the Triangular, by counting a layer of 1’s with Triangulars on top. When you get to 5, you have counted past Triangular.

I keep hearing 6 has too much ambiguity, and that means there’s inversion of SBE over a Counter. That generates a vastly larger Identity Space.

That reminds me. I heard today, while walking in the sleet along the river - it’s still 23 Fed 2023 - a Storyline completion in which Joana is able to explain how she began to think about George because she was playing with an idea that became Identity Spaces in which she would entangle two Things, meanings Ends, over a space in which stuff like group operations run. This would identify all the potential threads connecting the 2 Ends over that Identity Space.

There was a lot more, and more is flooding into me now, but I want to stick to the point, which is that I’ve never been able to say that before. I’ve never been able to say how and why they can connect. I saw then the obvious connection, which is that touch is deeply familiar. That’s 2 huge changes: the reason why and the reason why.

So, the reason Triangular has solutions is that you can treat it as a loop. Go over this. We can count Ends or Segments. Count the latter, then 1 is the ‘far’ End of the first Segment, that means 2 is the far End of the next, and thus 3 can return to 1 (or to anywhere, if you idealize, but always possibly to that 1, though located a 1-0Segment away in Winding. That is why there’s a general form.

By the same token, you can make the same loop with D2. You don’t have to count to the far End of that 1st Segment.

This is clearly correct, and it’s completely new thinking which fits to ideas as far back as Galois.

So, these are Regularizations, right? Remember, D2 doubles to D4 for grid squares, and that has to run through D3 as well. By run through, I mean that Pathway has to exist. Drawing in 1Space with 1Segments is simple: add or subtract a Segment and rearrange to an ideal shape. That’s all allowable in 1Space, not in 0Space. That’s why we see solutions which are those allowable in 0Space.

One thing that bothers me about this movie, which has been running in the background, is they made no effort to have the people look like the period. They look like modern actors with modern haircuts and attitudes, and often costumes. And speaking in modern ways.

————-

So, how does this work? A variation on the theme? Same old melody? Like I can list dimensions of why the US has become aggressive, and I can see them as Hexagons and thus as combinations of Hexagons in related layers, meaning I see attachments between layers and Extents within layers. I suppose an Extent is also a connection between layers, seen tipped that way. That’s neat: gives us Attachments and Extents as views!

So that makes part of the picture: local layers, connected by Attachments, and a larger map of local layers connected by Extents. So I could point to Eisenhower’s ‘military-industrial complex’, can count that over the years as an Extent, which allows development of the moral basis to which that ‘complex’ would naturally attach, together with the factors that developed into moral superiority, with that rooting in ‘democracy’, which is seen as the US mission and justification, etc. The idea is that these generate Pathways which come up to moments, to contexts.

Now that we have layers, we have fCM. That is true locally, because we make small decisions, and over various time frames.

[mathematicianadda]

Is there an area of Mathematics that was set back by the coming of computers?

From a book [1] I was reading: in the 19th century, Jacobi was able to compute highly accurate tables of elliptic integrals using his invented theta functions and inverse elliptic functions. The book suggests that had he and Legendre had a a computer, they would have used numerical integration to compile the tables, and would not have had the motivation to create the corresponding inverse functions and theta functions, which had led to a beautiful theory on their own. But also that on the other hand, a computer would have helped him determine that a part of that framework (the identities between functions) was not necessary either for what he needed them for...

It goes without saying that the benefits of computers to mathematics is tremendous in so many aspects, new highways have opened, and we are in a new era of thought because of it.

But that excerpt made me wonder:

What are we missing out on today in the field, because there is no longer a motivation to come up with ingenious ways to solve some such problems?

And the bonus question: what would the likes of Euler, Riemann, Gauss, and Galois have done if they had powerful computers at their fingertips? And what would they have not produced?

[1] "A = B" by Petkovsek, Wilf, Zeilberger

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[mathematicalrosebudthorn]

Embeddable category of embeddable objects is at least a 2-category: other categories are embeddable (groups, manifolds, etc); as you should keep embedding within higher Euclidean space and higher symmetric groups, the infinite versions of those respectively are the direct limits in these directed sets. To embed things, you need a notion of dimension/order; what else has such? Field extensions?

“A group almost everywhere” a measurable group: like a topological group but with measure such that the multiplication and inverse maps are measurable ae

Treating 2-categories like L2: is there some additional structure one could gleam from conjugacy relations 1/p + 1/q = 1?

Homology on homology groups (given by spectral sequences). On homotopy groups. Do these have a topology? How “close” is one algebraic structure to another? How “close” is a group to another group? What algebraic structures should be measured? Which ones preserved? When do two groups fail to be isomorphic? Can this be measured? (Presumably yes, using cohomology/derived functors or something). What other properties? How close is Z to Z/2? (In what sense, “close”?) What is group convergence ie what is a sequence of groups converging to another group (again in what sense? Spectral sequences?) Manifolds converge via Gromov-Hausdorff convergence. Do Lie groups converge to a Lie group in this sense? (Considering that limit may not even be a manifold (ie singularities) presumably not). What about other algebraic structures like categories? (if Gromov-Hausdorff metric is a metric on metric spaces, can we do something like a topology on topological spaces?)

Differential category theory; a differential category is one on which you can do analysis. Eg Lie groups are themselves differential categories as groups are categories. Differential algebra - analysis on algebraic structures, or with a d (see Josephi Ritt). Treat a category like a Banach space; finite rank/compact operator has finite dimensional image / apply to finite rank functor; figure out what objects get mapped to finite dim objects; figure out what objects have finite rank/torsion groups in the image.

Modulus of a curve family applied to homotopy classes?

Norm on modules; make long exact sequences into sequences of real numbers. Analysis <-> Algebra. Uncountable long exact sequences - just pairwise exactness. When are these induced in nature?

Higher homotopy are all functors; we should maybe consider (possibly continuous) families of functors. It seems the algebraic geometers are already here (deformation theory, moduli space, etc).

Galois functors = sheaves.

pi_infinity(X) = maps from S^infinity to X up to homotopy. (But S^infinity is contractible, so if X is path-connected, this is always 0. Perhaps you want a direct limit? But pi_k might commute with limits so then it’s still 0)

Excision analog for homotopy - it exists! Excision in the STABLE sense.

Necessary and sufficient conditions for torsion homotopy? See Serre 50s.

Banach bundle. Algebraic knots. Double tangent bundle; bubble cohomology; cohomology on the level of functors/categories. Symplectic homotopy; algebraically define symplectic structures.

Homotopy = weak; different notions of weak? Up to 0-homotopy, n-homotopy, etc? Up to rational q-homotopy? Etc

Hom from Galois extensions to Z (multiplicative extensions)

Meme = some categorical notion that transfers “techniques” from one field to another. How to formalize a “technique”?

Linear maps = structure preserving. What about nonlinear structures? Analysis/categories

Is there a way to determine if a constructive proof exists?

Metric functors - bifunctors from a category into R.

Using moduli spaces/categories to tackle differential equations. Is there a relationship between computation of groups and solving diff eqs? Are they equally as “difficult” (formally in some logic sense)? What does it mean to solve an equation? It means some relationship is satisfied; it means something exists. What does it mean to exist?

Sum of dimension of coho for complex >= 4 is related to general problem of computing actual coho groups / other functor image objects. Maybe try approaching this categorically? Rationally? Need analogous notion of rational homotopy except instead of just detecting torsion, also detects HOW much torsion -- or maybe, an easier question is, how many different TYPES of torsion (mod p1, p2, p3 ... ) p-adic (rational) homotopy; some type of object classifier (?) need way to translate problem about counting dimensions to counting torsion (or even something simpler, but a relationship between dimension and torsion would be nice as it also applies to spheres); maybe related to counting # of distinct roots (NOT including multiplicity) some kind of categorical argument principle; need some kind of categorical integral, like a functional functor. (IN FACT, people DO do this at least the first part; most of the stable stems have been computed via their p-primary components, so all the work is in detecting the different types of torsion, using p-primary parts. All the nontriviality is in p = 2, 3, and 5).

Are the orders of the stable stems bounded? We know they’re all finite. Spheres and primes? What’s the deal with the weird behavior at prime stable stems?

Categorical compact / connected? Continuous maps pullback open/closed sets and pushforward compact/connected sets. Aaron said this was related to topoi once.

Complex topology? Maybe there’s something more to complex manifolds topologically? K(C,n)? Complex homotopy? (This is deformation theory).

Dynamical systems approach to logic: change one initial condition (say, some axiom) and get drastically different results; already captured in categorical approach to dynamics as topoi form lattices which are graphs.

Categories and dynamical systems: dynamical systems give categories (arrows between objects, and so on) except with ADDITIONAL structure (like a metric or topology on the morphisms and the objects between them).

Moduli space of spaces of solutions to different parameterized diff eqs. Algebraic geometers already do this with polynomials, parametrized by their coefficients (a la deformation theory and variation of Hodge structures)

Hopf map Navier Stokes. 3D problems. 3D manifolds, low dim top. Spaces whose cohomology rings are fields.

If [X,-] is a functor from hTop to Grp then X is homotopic to the suspension of some space Z; NOT true. There exist cogroup objects that are not suspensions in hTop.

No satisfactory geometric description to this date (according to wiki) describing the abelian group structure for structure set S(X) in algebraic surgery exact sequence (a la Ranicki) (see surgery exact sequence on wiki)

(Algebraic) generalization of a flip. Need more algebraic notion of surgery to perform it on complex manifolds retaining the complex structure - find topological obstruction a la surgery exactness.

How do you make new complex/ac/sac manifolds WITHOUT algebraic geometry? You have a lot of them from varieties but these are all Kahler (induced from ambient projective space). There is a gap in our knowledge between algebraically obtaining complex manifolds and topologically doing so, as evident from integrability issues and the longstanding question of whether there exist almost complex but not complex manifolds above complex dim 3. Maybe use the ideas of Mori (flips) and Gompf (symplectic) and Eliashberg (contact) for a notion of surgery on varieties/complex manifolds that preserve the structure you want.

Almost complex cobordism. Algebraic Poincare conjecture (CPn). Poincare conjecture is a statement about the sphere spectrum in different categories (ie different morphisms). Birational cobordism? (Algebraic surgery already taken). Complex cobordism iff complex surgery? A la Milnor, as in the real case.

Decompactification to make a manifold not compact but still maintaining properties invariant wrt change in dimension (eg homotopy); push topology around dimensions (inter dimensional) so maybe we can hope to utilize low dim top techniques in high dim and vice versa. Probably barely any properties are preserved in this.

Chain surgery; encode surgery through algebra on chain level? Dennis thinks all the action for topology is now entirely on the chain level; DGAs and all that captures all the homotopy information. 

[dnsiaody]

CSIR UGC NET Application 2017

COMMON SYLLABUS FOR PART ‘B’ PLUS ‘C’ MATHEMATICAL SCIENCES DEVICE - 1 Analysis: Primary set theory, finite, countable and uncountable sets, True number system as the complete ordered field, Archimedean property, supremum, infimum. admission.scholarshipbag.com Sequences and series, convergence, limsup, liminf. Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, uniform continuity, differentiability, indicate value theorem. Sequences plus a number of functions, homogeneous convergence. Riemann sums plus Riemann integral, Improper Integrals. Monotonic functions, types associated with discontinuity, functions of bounded variation, Lebesgue measure, Lebesgue integral. Functions of various variables, directional derivative, partially derivative, derivative as being a geradlinig transformation, inverse and implied function theorems. Metric areas, compactness, connectedness. Normed geradlinig Spaces. Spaces of constant functions as examples. Geradlinig Algebra: Vector spaces, subspaces, linear dependence, basis, aspect, algebra of linear changes. Algebra of matrices, position and determinant of matrices, linear equations. Eigenvalues plus eigenvectors, Cayley-Hamilton theorem. Matrix representation of linear changes. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms. Internal product spaces, orthonormal base. Quadratic forms, reduction plus classification of quadratic types UNIT - two Complicated Analysis: Algebra of complicated numbers, the complex aircraft, polynomials, power series, transcendental functions such as rapid, trigonometric and hyperbolic features. Analytic functions, Cauchy-Riemann equations. Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus rule, Schwarz lemma, Open umschlüsselung theorem. Taylor series, Laurent series, calculus of residues. Conformal mappings, Mobius changes. Algebra: Permutations, combinations, pigeon-hole principle, inclusion-exclusion principle, derangements. Fundamental theorem of math, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s Ø- function, primitive origins. Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups, Cayley’s theorem, class equations, Sylow theorems. Rings, ideals, excellent and maximal ideals, quotient rings, unique factorization site, principal ideal domain, Euclidean domain. Polynomial rings plus irreducibility criteria. Fields, limited fields, field extensions, Galois Theory. Topology: basis, thick sets, subspace and item topology, separation axioms, connectedness and compactness. UNIT -- 3 Ordinary Differential Equations (ODEs): Existence and originality of solutions of preliminary value problems for initial order ordinary differential equations, singular solutions of initial order ODEs, a system associated with first order ODEs. The common theory of homogenous plus nonhomogeneous linear ODEs, deviation of parameters, Sturm-Liouville border value problem, Green’s functionality. Partial Differential Equations (PDEs): Lagrange and Charpit strategies for solving first purchase PDEs, Cauchy problem intended for first order PDEs. Category of second order PDEs, General solution of increased order PDEs with continuous coefficients, Method of splitting up of variables for Laplace, Heat and Wave equations. Numerical Analysis: Numerical options of algebraic equations, Technique of iteration and Newton-Raphson method, Rate of convergence, Solution of systems associated with linear algebraic equations making use of Gauss elimination and Gauss-Seidel methods, Finite differences, Lagrange, Hermite and spline interpolation, Numerical differentiation and incorporation, Numerical solutions of ODEs using Picard, Euler, customised Euler andRunge-Kutta methods. Calculus of Variations: A variety associated with a functional, Euler-Lagrange formula, Necessary and sufficient situations for extreme. Variational strategies for boundary value troubles in ordinary and partially differential equations. Linear Essential Equations: The Linear integral formula of the first plus second kind of Fredholm and Volterra type, Options with separable kernels. Feature numbers and eigenfunctions, resolvent kernel. Classical Mechanics: General coordinates, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s rule and the principle of minimum action, Two-dimensional motion associated with rigid bodies, Euler’s dynamical equations for the movement of the rigid entire body about an axis, the concept of small oscillations. DEVICE - four Descriptive figures, exploratory data analysis Example space, discrete probability, 3rd party events, Bayes theorem. Unique variables and distribution features (univariate and multivariate); requirement and moments. Independent unique variables, marginal and conditional distributions. Characteristic functions. Possibility inequalities (Tchebyshef, Markov, Jensen). Modes of convergence weakened and strong laws associated with large numbers, Central Restrict theorems (i. i. g. case). Markov chains along with finite and countable condition space, classification of claims, limiting behaviour of n-step transition probabilities, stationary submission, Poisson and birth-and-death procedures. Standard discrete and constant univariate distributions. sampling distributions, standard errors and asymptotic distributions, distribution of purchase statistics and range. Strategies of estimation, properties associated with estimators, confidence intervals. Testing of hypotheses: most effective and uniformly most effective tests, likelihood ratio testing. Analysis of discrete information and chi-square test associated with goodness of fit. Big sample tests. Simple non-parametric tests for just one particular and two sample troubles, rank correlation and check for independence. Elementary Bayesian inference. Gauss-Markov models, estimability of parameters, best geradlinig unbiased estimators, confidence times, tests for linear ideas. Analysis of variance plus covariance. Fixed, random plus mixed effects models. Assured multiple linear regression. Primary regression diagnostics. Logistic regression. Multivariate normal distribution, Wishart distribution and their qualities. Distribution of quadratic types. Inference for parameters, partially and multiple correlation coefficients and related tests. Information reduction techniques: Principle element analysis, Discriminant analysis, Bunch analysis, Canonical correlation. Easy random sampling, stratified sample and systematic sampling. Possibility proportional to size sample. Ratio and regression strategies. Completely randomised designs, randomised block designs and Latin-square designs. Connectedness and orthogonality of block designs, BIBD. 2K factorial experiments: confounding and construction. Hazard functionality and failure rates, censoring and life testing, collection and parallel systems. Geradlinig programming problem, simple strategies, duality. Elementary queuing plus inventory models. Steady-state options of Markovian queuing versions: M/M/1, M/M/1 with restricted waiting space, M/M/C, M/M/C with the limited waiting area, M/G/1. All students are usually required to answer queries from Unit I. College students in mathematics are anticipated to answer an additional query from Unit II plus III. Students with within statistics are required in order to answer the additional question through Unit IV.

[mathematicianadda]

Question about inertia groups and unramified extensions

Let $K$ be a number field, and $v$ a finite place. If $\bar{K}$ is a separable closure of $K$, then in $G_K=\text{Gal}(\bar{K}/K)$ we can find the decomposition group of (a place over) $v$, which is isomorphic to the Galois group of $\bar{K_v}/K_v$, with $K_v$ the completion at $v$.

It is well know that the fixed field of the inertia $I_v$ in $\bar{K_v}$ is the maximal unramified extension of $K_v$. Is it also true that the fixed field of the inertia in $\bar{K}$ is the maximal extension of $K$ unramified at $v$? I think this is true, since we can easily move to the finite case where is true, but also I could appreciate a check.

After this, if we consider the maximal extension of $K$ unramified at $v$ and $v'$, with $v\ne v'$, then it is the intersection of the maximal extension unramified at $v$ with the one unramified at $v'$ (it is true? It seems to me obvious), therefore, by Galois correspondence, the product of the inertia $I_vI_{v'}$ is the group corresponding to that field.

But what happens if we consider the maximal extension unramified outside a finite set of places, so unramified at an infinite set of places? The infinite intersection would correspond to an infinite product of subgroups, which of course make no sense. So have we to compute it with, maybe, inverse limit, or something like this?

(My final goal is to understand a proof in Rubin's book Euler systems: he proved that, given a Galois representation $T$ with coefficients in the valuation ring $O$ of a finite extension of $\mathbb{Q}_p$, and a finite set of primes $\Sigma$ containing all primes where $T$ ramifies, primes above $p$ and infinite places, then the Selmer group $S^{\Sigma}(K,T)$ is equal to $H^1(K_{\Sigma}/K,T)$, where $K_{\Sigma}$ is the maximal extension unramified outside $\Sigma$.

The proof is the following: $$\begin{split}S^{\Sigma}(K,T)&\overset{(1)}{=}\ker \left(H^1(K,T)\to \prod_{v\not\in\Sigma} H^1(K_vT)/H_f^1(K_v,T)\right)= \\&\overset{(2)}{=}\ker\left(H^1(K,T)\to\prod_{v\not\in\Sigma}\text{Hom}(I_v,T)\right)=\\&\overset{(3)}{=}\ker\left(H^1(K,T)\to H^1(K_{\Sigma},T)\right){=}H^1(K_{\Sigma}/K,T). \end{split}$$ (1) is the definition. I think that in (2) we need $\text{Hom}(I_v,T)^{Fr}$, the fixed points of Frobenius, so I cannot understand that passage. But my big problem is in (3), with which the question is related).

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