A Counterexample to Kurosh’s Division-Ring Problem

1 Introduction

A division ring D is algebraic over its center F=Z(D) if every a\in D satisfies a nonzero polynomial in F[t]. For a subset S\subseteq D, write F(S) for the division subring generated by F\cup S. The ring is locally finite if

The Kurosh problem for division rings asks whether algebraicity over the center implies local finiteness. We give a negative answer.

All rings and subrings in this paper have the same identity. If k is a central subfield and x,y belong to a ring, k[x,y] denotes the unital k-algebra generated by x,y; the generators need not commute. The free polynomial algebra on two formal generators is denoted k\langle x,y\rangle.

The same pair \{x,y\} therefore witnesses the failure of local finiteness. The construction does not prescribe the center. It first produces an algebraic division ring over a countable central subfield, and a separate finite-descent argument proves infinite dimension over the actual center. The annihilating polynomial is allowed to depend on the element.

1.1 Historical context

Kurosh formulated his question in 1941 in connection with Burnside’s finiteness problem for groups. He asked whether a division ring that is algebraic and finitely generated as an algebra over a central subfield must be finite-dimensional over that subfield [12, Problem (K), p. 240]. The subfield in this formulation need not be the full center, and finite generation means algebra generation, not merely division generation.

The broader problem for algebraic algebras has a different history. Golod’s construction, based on the Golod–Shafarevich estimates, gives finitely generated infinite-dimensional nil algebras [5, Example 1]. Their unitizations are algebraic algebras with nonzero nilpotents, so they do not answer the division-ring question.

There are positive results under additional hypotheses. Jacobson’s bounded-degree theorem gives a polynomial identity, and Kaplansky’s theorem makes algebraic algebras satisfying a polynomial identity locally finite; see [7, Theorems 3 and 5]. Bell, Drensky, and Sharifi later proved that a division ring whose elements satisfy nonzero polynomial relations of uniformly bounded degree d, with coefficients on the left in a possibly noncentral subfield, has dimension at most d^2 over its center [2, Theorem 1.3]. Our construction imposes element-dependent relations without a uniform degree bound.

Amitsur proved that a division algebra finitely generated as an algebra over an uncountable central field is automatically algebraic [1, Corollary 2']. This conclusion is algebraicity, not local finiteness. Countability plays a different role here: it allows us to schedule all polynomial expressions in the construction. Goodearl and Letzter’s recent work [6] treats nonfinite generation as an algebra for classes of transcendental skew fields and explicitly distinguishes that question from the algebraic Kurosh problem. Theorem 1.1 concerns algebraic division rings and their actual centers.

The proof combines standard division-algebra constructions with matrix and formal-series calculations. Symbol and cyclic algebras provide division-ring models for the generic block data [4, Section 2.5]. Traces organize cyclic word relations [16, Section 4], while inner derivations describe infinitesimal conjugation [8, Theorem 8]. The lifting argument supplies the additional response, compatibility, and generation statements needed to preserve two independent deformation directions while imposing successive algebraic relations.

1.2 The construction

The main ingredient is Theorem 2.1. Its input is a tuple of formal series in two noncommuting variables over a finite-dimensional central division algebra. The constant tuple generates the coefficient algebra, and the two linear coefficients are independent modulo changes coming from conjugation. The theorem makes any prescribed series algebraic over a larger countable central field. At the same time it increases the degree of the division algebra, preserves all previous polynomial identities, and restores both generation and the two independent directions. These last requirements allow the construction to continue.

For the prescribed series w, the immediate goal is to find a formal substitution Z=(Z_1,Z_2) for which w(Z) satisfies the minimal polynomial of the chosen constant output w(Z(0)). The input Z(0) is a nonzero generic input, small with respect to an auxiliary valuation, not the original zero input. After splitting the coefficient algebra into matrices, we take the constant input to be a diagonal sum of independent generic matrix pairs. Varying a diagonal block gives a first eigenvalue response; adding an arrow to another block and an arrow back gives a paired response. Sections 4 and 5 establish the formal evaluation rules and show that these responses partition the eigenvalue labels into linked groups.

A transfer theorem puts each first unsolved noncommutative word coefficient in the same response space. It also turns proportional responses into functional relations between eigenvalue branches (Sections 6 and 7). The relations within each block then separate from a fixed-dimensional space of relations among blocks. To test the latter, we repeat each block and attach auxiliary blocks, called hubs. Individual hubs distinguish repeat positions; a common hub compares them. This matrix test proves that every leading error satisfies the required cyclic sums of word coefficients (Sections 8 and 9).

An eleven-block arrangement of two colors supplies corrections whose only obstructions are those cyclic sums. We solve the equations one word length at a time in Section 10. A cyclic division algebra realizes these matrix calculations over a division ring, and a Taylor translation restores generation and the two independent directions (Sections 11 and 12). The corrections are solutions of finite affine linear systems; this is what permits their descent from a splitting field to the division algebra’s center. If w already satisfies its constant term’s minimal polynomial, the spectral correction is unnecessary, but realization and translation are still needed to increase degree and restore the iteration hypotheses.

Section 3 first explains how the lifting theorem yields Theorem 1.1. The fields introduced during the iteration are countable, so every polynomial expression can be scheduled. A division-ring ultraproduct retains the resulting relations. Its two-generated subalgebra is already a division ring, and finite word sets prove that its dimension over the central coefficient field is infinite. The actual-center argument concludes this deduction. The remaining sections prove the lifting theorem and all its required compatibilities.

2 The lifting theorem

The construction proceeds through finite-dimensional central division algebras. At each step we impose an algebraic relation on one prescribed element while preserving the freedom to enlarge the algebra at the next step. Formal power series in two noncommuting variables record that freedom.

For a k-algebra A, write

for the ring of formal series in words in z_1,z_2, with coefficients in A commuting with the variables. Multiplication concatenates words and multiplies their coefficients in their original order. The ring is complete for the filtration by word length. For a tuple X=(X^{(1)},\ldots,X^{(u)}) of such series, let X_0\in A^u be its constant tuple and let X_1,X_2\in A^u be its coefficients of the two individual letters. Thus the subscripts 1,2 denote letters, not degrees. Set

The classes of X_1,X_2 modulo this subspace describe changes of the tuple that cannot be obtained by infinitesimal conjugation.

Theorem 2.1 is the central construction. Its application to the Kurosh problem uses u=2 and addresses one polynomial in the two generators at a time. The fields are allowed to grow; they need not be algebraic extensions of their predecessors. The final step will therefore identify the actual center separately and prove infinite dimension over it.

Figure 1 separates the two tasks within a lifting step. The response calculation makes the prescribed series algebraic; the final translation restores the hypotheses needed to repeat the step. The two transverse directions must survive together, not merely one at a time.

3 The countable construction and the actual center

We now apply Theorem 2.1 repeatedly. Recall its full conclusion: from a finite-dimensional central division algebra A/k and a tuple of free series X whose constant term generates A and whose two linear coefficients are independent modulo inner directions, it produces such a tuple over a countable extension k' and a central division algebra of strictly larger degree. It preserves every old polynomial identity and gives a monic algebraic relation for any chosen series w. The countability, generation, and two-direction conclusions are all needed to iterate this theorem.

The resulting division algebras need not embed in one another. We use an ultraproduct to retain their polynomial relations, and then prove separately that the resulting algebra has infinite dimension over its actual center.

3.1 Scheduling all polynomial expressions

Start with

Its constant pair (0,0) generates k_0 as a unital k_0-algebra. The inner-direction space is zero, and its two linear coefficients (1,0) and (0,1) are independent. Thus it satisfies the hypotheses of Theorem 2.1.

Inductively we obtain countable fields

finite-dimensional central division algebras A_i/k_i of strictly increasing degrees, and pairs

satisfying the same generation and transverse-independence hypotheses. Write (x_i,y_i)=X^{(i)}_0 for the constant pair; in particular,

Here is a schedule that ensures algebraicity of every expression needed in the limit. When k_i has been constructed, enumerate the countable set k_i\langle x,y\rangle as (p_{i,j})_{j\geq0}. Enumerate the pairs (i,j)\in\mathbb N^2 by increasing i+j, in a fixed order within each diagonal. If (i_n,j_n) is the nth pair, beginning at n=0, then i_n\leq n. At stage n, apply the lifting theorem to

This expression is defined over k_n, since k_{i_n}\subseteq k_n. The identity w-p_{i_n,j_n}(X^{(n)})=0 is preserved, so the new relation for the lifted w gives a nonzero monic polynomial

such that

Every later lifting preserves this polynomial identity, now regarded over its coefficient field k_{n+1}. Taking constant terms shows that the corresponding identity holds at (x_h,y_h) for every h\geq n+1. Each polynomial over each k_i is treated at a finite stage.

3.2 A two-generated algebraic division ring

Fix a nonprincipal ultrafilter \mathcal U on \mathbb N and form (see [10] for the general construction)

Here two sequences represent the same element when their equality set belongs to \mathcal U. This is a division ring: a nonzero class has nonzero coordinates on a set in \mathcal U, and taking their coordinatewise inverses on that set gives its inverse.

Put k_\infty=\bigcup_i k_i. An element c\in k_j determines a class in \mathcal D by taking its image in the central field k_i\subset A_i for i\geq j and choosing arbitrary coordinates at the finitely many earlier indices. These classes are independent of the early choices and of j. They define a central field embedding

the field operations agree at every sufficiently large index, and a nonzero field element stays nonzero at all those indices. This construction uses only the inclusions of coefficient fields; it does not require embeddings between the algebras A_i.

Let

Every r\in R has the form p(x,y) for some p\in k_j\langle x,y\rangle. The schedule and (3.2) provide a nonzero monic polynomial q\in k_h[t], for a suitable h, such that q(p(x_i,y_i))=0 at every sufficiently large index. Therefore q(r)=0 in \mathcal D. Every element of R is algebraic over k_\infty.

This already makes R a division ring. For 0\ne r\in R, choose a nonzero relation of least degree

Its constant coefficient is nonzero: otherwise multiplication by r^{-1} in the ambient division ring \mathcal D would give a shorter relation. Consequently

Since k_\infty is countable and R is generated by two elements as an algebra, R is countable. Its characteristic is zero.

3.3 Infinite dimension and passage to the center

We first prove

For each i and \ell\geq0, let V_{i,\ell} be the k_i-span of the words in x_i,y_i of length at most \ell, including the empty word 1. If V_{i,\ell}=V_{i,\ell+1}, then V_{i,\ell} is stable under right multiplication by x_i and y_i and contains every word. By (3.1), it equals A_i. Until this happens, the dimensions increase strictly, starting from 1.

Fix \ell. The dimensions [A_i:k_i] tend to infinity, so

for every sufficiently large i. There are only finitely many subsets of size \ell+1 in the finite list of words of length at most \ell. The ultrafilter therefore selects one fixed subset w_0,\ldots,w_\ell that is linearly independent at a set of stages belonging to \mathcal U.

These same words are independent over the embedded k_\infty in R. Indeed, the coefficients of a putative relation

all belong to one field k_h. Equality in the ultraproduct holds on a set in \mathcal U. Intersecting it with the independence set and the tail i\geq h gives a nonempty set where stage independence forces every c_j to vanish. Thus R has arbitrarily large finite linearly independent sets, proving (3.3).

Finally let F=Z(R) be the actual center. It contains k_\infty and is algebraic over k_\infty, since every element of R is. In particular, every element of R is algebraic over F. Also

because F\subseteq R and R is already a division ring.

We use finite descent of the multiplication table and generator coordinates; compare the proof of the local-finiteness criterion in [3, Section 4]. Suppose, for a contradiction, that [R:F]=n<\infty. Choose an F-basis b_1=1,b_2,\ldots,b_n of R, and write

with all coefficients in F. Let E be the subfield of F generated over k_\infty by these finitely many coefficients. Algebraicity gives [E:k_\infty]<\infty. The space

contains 1,x,y and is closed under multiplication by the displayed structure constants. It is therefore a k_\infty-subalgebra of R containing k_\infty[x,y]=R, so B=R. This implies

contrary to (3.3).

Thus the countable characteristic-zero division ring D=R is algebraic over its actual center F, while

The finite set \{x,y\} witnesses the failure of local finiteness.

This proves Theorem 1.1, subject only to Theorem 2.1, which we now prove.

4 Formal evaluations and generic spectra

We begin the proof of Theorem 2.1 by constructing a substitution Z=(Z_1,Z_2) for the two free letters. Its coefficients will commute with the old algebra A, so applying it to (X,w) preserves their polynomial identities. The matrix calculation focuses on w; Sections 11 and 12 realize the substitution in a division algebra of larger degree and restore generation and both transverse directions for X(Z).

Throughout the lifting argument, k is a field of characteristic zero containing all roots of unity, A is a central division algebra of degree d over k, and

The letters commute with A, but not with each other. Thus an element is a family (a_\omega)_\omega indexed by finite words, written \sum_\omega a_\omega z^\omega, and multiplication uses concatenation of words. Fix an algebraic closure K of k and a splitting A\otimes_k K\simeq M_d(K); this uses the splitting and scalar-extension properties of central simple algebras [14, Chapter IV, Example 2.14(a) and Proposition 2.15].

If P=(P_1,P_2) is a pair of n\times n indeterminate matrices, put

Here K[[P]] is the ordinary commutative power series ring in the 2n^2 entries. We omit the subscript n when the size is clear. The coefficient matrices act on the first tensor factor. In particular, every input arrow between blocks is a pair of scalar matrices acting on the second factor. It is not an arbitrary pair of matrices on both factors.

In the spectral case we seek a substitution Z=(Z_1,Z_2), with constant term R, for which W(Z) is conjugate to a split semisimple constant output W(R). If \Lambda is the set of distinct eigenvalues of W(R), this gives

The division realization will identify q with the minimal polynomial over its central coefficient field. If w already satisfies the minimal polynomial of w(0), substitution retains that relation and the spectral correction is unnecessary.

For the spectral construction, we will find matrix sizes with a common eigenvalue multiplicity, disjoint spectra at independent inputs, and scalar spectral blocks under commuting Taylor perturbations. These properties will make the eigenvalue responses of Section 5 well-defined. We first specify the evaluations and specializations used to establish them and to evaluate triangular auxiliary inputs later.

4.1 Convergence and specialization

Let E be a complete discretely valued field containing K, with valuation v, such that v(c)=0 for c\in K^\times. We use E=C((\varepsilon)), or a finite extension of such a field, and normalize v(\varepsilon)=1. Matrix valuations mean the minimum of the entry valuations. A matrix is small if this minimum is positive. A small generic point is P=\varepsilon P^\circ, where all entries of P^\circ are algebraically independent over the previously specified constant field. Several independent points can use the same \varepsilon.

The finite extensions used to split spectra are separable in characteristic zero. The valuation extends to each such extension, which remains complete and discretely valued [13, Theorem 7.38 and Remark 7.43]; with the normalization v(\varepsilon)=1, its value group may be \frac1e\mathbb Z for a positive integer e.

We will use inverses of finite matrices as rational operations after evaluation. The following elementary distinction is important. Although (\varepsilon-Q)^{-1} has a formal expansion at Q=0 over E, that expansion need not converge after Q=\varepsilon Q^\circ. Accordingly, when transferring an identity involving an inverse, we first multiply by its determinant. We evaluate the resulting numerator by Lemma 4.1, and divide only when the evaluated determinant is nonzero. Formal inverse expansions are used to define coefficients; they are not presumed to converge at every subsequent specialization.

In applications, the coefficients of a cleared numerator are finite sums of products of base-point scalars and matrix-entry series. The path estimate gives a uniform valuation bound in the auxiliary coordinates. Lemma 4.2 therefore identifies formal identities in these coordinates with identities at independent small generic points. The base-point valued field is constructed first; the auxiliary residues are adjoined afterward. This order keeps their independence valid even after the base spectrum has been split. The argument can be applied separately at each Taylor degree.

We record two further field conventions. Evaluation

is injective, by homogeneous degree or by Lemma 4.2. It extends to the fraction field, and a finite splitting field embeds into a finite algebraic extension of the target. Also, if R is a small constant pair, then

is injective: reduction modulo the valuation ideal sends the image of f to f(\Delta). We use this observation for small block diagonal R. Denominators in the fraction field remain nonzero, so the map extends to fraction fields.

Finally, the Taylor substitution P\mapsto P+\eta v maps \operatorname{Frac}K[[P]] into \operatorname{Frac}K[[P]][[\eta]], with reduction the identity. It extends to a finite separable extension \Omega with values in \Omega[[\eta]]: choose a primitive element and lift its defining equation using its nonzero derivative. The lift reducing to the chosen primitive element is unique, by coefficient recursion. This convention lets us move a base point while retaining its labeled eigenvalue branches.

4.2 Generic division evaluations

Generic matrix evaluations have a useful divisibility property for their eigenvalue multiplicities. We obtain it by first evaluating inside a division algebra.

For n\ge1, choose independent central variables \alpha,\beta and a primitive nth root of unity \zeta_n\in k. Over k_0=k(\alpha,\beta) let B_n be generated by x,y with

For n=1 this means B_1=k_0. The reduced monomials x^iy^j, 0\le i,j<n, form a basis. Indeed they span by the relations, and after adjoining nth roots of \alpha and \beta their clock-and-shift matrices are linearly independent. These matrices span M_n, so B_n is central simple of degree n. This is the standard symbol construction [4, Corollary 2.5.5(1)], with the inverse root of unity for our convention.

Choose a basis b_1,\ldots,b_{n^2} of B_n, and let c_{ij}, i=1,2, j=1,\ldots,n^2, be independent variables. Evaluate w at

This gives an element of a central division algebra of degree dn over

After splitting A and B_n by constants independent of the c_{ij}, the b_j become a basis of M_n. The change from the c_{ij} to the entries of P_i/\varepsilon is therefore an invertible linear change of independent coordinates. The resulting evaluation is a small generic scalar matrix evaluation in the sense fixed above.

Let e_n denote the minimal-polynomial degree of this element. Its minimal polynomial is irreducible and separable over F_n. Consequently W(P) is semisimple over a splitting field, and every distinct root has the same multiplicity

For completeness, if D/F has degree N and a\in D has minimal polynomial q of degree e, then left multiplication by a has characteristic polynomial q^{N^2/e}: view D as a vector space over F[a]. After splitting D=M_N, left multiplication acts by the matrix a on each of its N columns. Its characteristic polynomial is therefore the Nth power of the matrix characteristic polynomial. Comparing root multiplicities gives N/e in the matrix representation. This also proves that N/e is an integer.

Linear independence of 1,a,\ldots,a^{e-1} is preserved by field extension. Thus the same e_n,t_n describe W(P) over a splitting field of \operatorname{Frac}K[[P]].

4.3 Nonconstant roots and stable multiplicity

Let q_0 be the minimal polynomial of w(0) over k. If q_0(w)=0 as a free series, the subsequent spectral construction is unnecessary; the unconstrained case of the lifting argument will use this existing relation. In the rest of this section assume q_0(w)\ne0.

A nonzero homogeneous part of q_0(w) is detected by scalar matrices of every sufficiently large size. To see this, choose a word of length L with nonzero coefficient and use a path of L+1 vertices, placing its successive letters on the consecutive matrix-unit arrows. The (1,L+1) entry detects precisely that coefficient. Padding by zero blocks works for all larger sizes. It follows that q_0(W(P))\ne0 at every generic size n\ge n_0, for some n_0.

Choose p\ge n_0 for which the positive integer t_p is minimal among all t_n, n\ge n_0, and put

We henceforth use sizes in \mathcal S. At size n, the minimal polynomial always has degree at most e_n=dn/t at every admissible specialization. The following consequence controls all commuting Taylor deformations used below.

In particular, the roots at a small semisimple point with e_n distinct roots have labeled scalar Taylor branches. The splitting field of the generic point embeds in their Taylor fraction field by (4.1); a labeling is chosen by reduction to the roots at the base.

Let \rho_n be the generic rank of the input Jacobian of

Its minors are formal entry identities, so \rho_n bounds the corresponding specialized ranks. At a maximal-root-count point the scalar branches satisfy

The first e_n rows give an invertible Vandermonde matrix. Hence the trace Jacobian and the eigenvalue Jacobian have the same rank there, including over a commuting Taylor fraction field. We have therefore obtained both ingredients needed for the response calculations: scalar branches and specialization-stable rank bounds.

5 The space of eigenvalue responses

Fix a size n\in\mathcal S, a generic input pair P, and a splitting field \Omega_P of W(P) over \operatorname{Frac}K[[P]]. Write its distinct roots as \lambda_h, with spectral projections \pi_h, 1\le h\le e_n. Each projection has rank t. We also use these data after embedding into the valued fields of Section 4.

We seek the possible leading changes of the eigenvalues at P. Besides varying P itself, we allow an excursion through another input block and a return to P. Their linear span will determine which eigenvalues can be varied independently.

5.1 Loop and paired responses

Upper triangular block evaluations are also the basic mechanism of noncommutative difference-differential calculus; see [9, Proposition 2.2 and Theorem 3.11]. Here the admissible evaluations of Section 4 let us extract scalar eigenvalue responses from those block calculations.

For a perturbation \mathcal W of W(P), put \pi=\pi_h, \bar\pi=1-\pi, and

This operator acts on the complementary space. Its constant term is invertible, so its inverse has a unique expansion in either commuting or free formal parameters. Define the fixed-root Schur expression by

This expression records the obstruction to keeping the root \lambda_h fixed. For u in the image of \pi, the vector

is the unique vector projecting to u whose complementary component under \mathcal W-\lambda_h vanishes. Its remaining component is S_h(\mathcal W)u. Thus S_h(\mathcal W)=0 means that the fixed-root kernel is a graph over the original eigenspace.

When the constant output is split semisimple and these equations are imposed for every one of its distinct roots, the kernel graphs combine into a formal change of basis. The output remains conjugate to its constant term and satisfies that constant term’s minimal polynomial. Lemma 10.2 proves the equivalence with the polynomial identity at every finite word order.

The same Schur expression is defined near \operatorname{diag}(P,C_0) if \lambda_h is absent from W(C_0), embedding \pi_h at the P block. In this use, equations indexed by labels at P record those fixed roots only.

For a loop direction Y=(Y_1,Y_2) at P, define the scalar D_h(P;Y) by

This compression is scalar: apply Lemma 4.6 to the commuting deformation P+xY and take its first coefficient. Equivalently, D_h(P;Y) is the differential of the branch \lambda_h in direction Y. By Lemma 4.4, each of these linear functionals is nonzero. Because K is infinite, we can choose a constant direction v over K with

Indeed the forbidden directions form a finite union of proper K-linear subspaces, even though their defining coefficients belong to \Omega_P.

Let Q be an independent generic point of size n'\in\mathcal S. Let U be an input arrow P\leftarrow Q and V an input arrow Q\leftarrow P; each is a pair of rectangular scalar matrices. Form the triangular input pair

Set

Block paths show that this is bilinear in U,V. The inverse exists because the spectra at independent points are disjoint.

5.2 Coefficient spans and their specializations

At Q=0 the matrices W(Q)-\lambda_h are invertible, since no \lambda_h belongs to the spectrum of w(0). Hence (5.4) has a formal expansion in the entries of Q over \Omega_P. Define

Here D=D(P;\cdot):M_n(\Omega_P)^2\to\Omega_P^{e_n} is the linear map of loop responses. Its columns are its values on the standard coordinate directions in the two input matrices. The scalar identity of Lemma 5.1 holds also for these formal expansions, by clearing their determinants and applying Lemma 4.2.

Define \mathcal T_P\subseteq\Omega_P^{e_n} to be the linear span of the columns of D and of every coefficient vector of m, for every allowed partner size n'\in\mathcal S and every pair of coordinate arrows U,V. Write

Although the definition permits infinitely many partner sizes and coefficients, the ambient space is finite dimensional.

The lemma also applies after adjoining Taylor or Laurent parameters. One first clears determinants and then works coefficientwise as in Lemma 4.1. In particular, it permits triangular partners with repeated base blocks, provided the inverses required for the selected target roots exist in the chosen Laurent field.

5.3 A partition of the eigenvalue labels

The active space has a stronger structure than a general linear subspace. Its coordinates form groups whose responses are always in a fixed ratio.

We call two labels in the same part linked. The number of parts is a_n. Selecting one representative from each part identifies \mathcal T_P with the full coordinate space on those representatives. The next stage must show that these response relations control higher-order free-letter errors, rather than only the two elementary perturbations used to define them.

6 Transfer of the first unsolved word coefficient

Our next objective is to control equations in free letters. The first and paired responses describe matrix perturbations with one loop or two opposite arrows. We show that they also contain the first unsolved coefficient of an arbitrary free-series perturbation. The argument unfolds a word along a triangular chain and then replaces its internal states by a single auxiliary partner. The only difficulty is that this partner initially has the target eigenvalue. The following linear-algebra observation explains how to remove that singularity without changing the coefficient being computed.

We now recall exactly which response identities enter the transfer argument. At a small generic point P of size n\in\mathcal S, the eigenvalues \lambda_h have projections \pi_h of rank t. The first response is D_h(P;Y)\pi_h. A pair of arrows through an independent partner Q gives m_h(P,Q;U,V)\pi_h. The columns of the first responses and the coefficient vectors of all these paired responses span \mathcal T_P. By Lemma 5.2, their scalarity and all linear containment identities persist for a block-triangular partner after its finitely many resolvent determinants are cleared. For a set of target labels I, only the determinants belonging to labels in I are needed. We use the notation

These spaces and identities are extended to the coefficient field in use. We also fix a direction v with D_h(P;v)\ne0 for every target label.

Proposition 6.2 is the bridge from responses to free-series equations. It controls a first unsolved coefficient without assuming that the perturbation is commutative or that the auxiliary matrix has a separated spectrum before regularization. We next use its commuting-parameter consequence to show that linked responses arise from actual formal relations between eigenvalues.

7 Functional linkage and replication

We now show that the dimension of the active response space is additive under generic direct sums. The main point is to turn the linear relations between linked responses into identities between eigenvalue branches. These identities will continue to control responses through partners of every allowed size.

Retain the progression \mathcal S=\{pN:N\geq 1\} and the common eigenvalue multiplicity t. Thus, at size n\in\mathcal S, the number of distinct generic eigenvalues is e_n=dn/t. We use the following previously established facts. At a direct sum of independent small generic inputs of sizes in \mathcal S, the spectra are disjoint and have the maximal number of distinct eigenvalues. Their commuting Taylor deformations therefore have scalar spectral blocks, by Lemma 4.6. The response identities admit the specializations of Lemma 5.2, and their leading terms satisfy Proposition 6.2.

For a generic input P, let \Omega_P be its spectral splitting field, and choose a constant input direction v such that D_h(P;v)\neq 0 for every eigenvalue label h. Recall the active-space partition \mathcal P_P, characterized by

Labels in one member of this partition are called linked. In particular, a_n=\dim_{\Omega_P}\mathcal T_P is the number of its members. The argument below uses the full coefficientwise definition of \mathcal T_P: it contains the loop responses D(P;Y) and the coefficient vectors of m(P,Q;U,V) for every partner size in \mathcal S.

7.1 A formal identity for linked branches

7.2 All partner sizes and the dimension formula

We next pass from an identity near a direct sum to an identity for every coefficient defining the generic active space. This step is necessary because a bound proved using a fixed collection of partners would not bound a_n.

7.3 Partners of one common size

The dimension formula permits us to use a fixed matrix size on both ends of every paired response. We will also need to fix one arrow and vary only its reverse.

8 The common space of spectral relations

The active space describes the possible responses at one block. We now identify the relations that remain when many independent blocks are used together. The essential point is that the derivatives at different triangular inputs belong to one common space. A bound on their ranks separately would not give the relations needed for the free-series construction.

We retain the preceding notation. The allowed sizes are \mathcal S=\{pN:N\geq1\}; at size n\in\mathcal S, there are e_n=dn/t distinct generic roots, each of multiplicity t. The active space has dimension a_n, and \rho_n is the generic rank of the eigenvalue differential, equivalently of the trace differential. In particular \rho_n\leq a_n. We use the replication identity a_{pN}=Na_p and the following consequence of the preceding attainability argument. There are a size r\in\mathcal S and an integer K_0\geq1 such that the loop responses at a generic size-r point, together with paired responses through K_0 independent generic size-r partners, span its active space. In each pairing, either arrow may be fixed generically and the opposite arrow varied. All spaces below are extended to a common splitting field when several independent points are involved.

Fix b,a,g as in Lemma 8.2. For each generic size-b point P, let E_P be the a-dimensional space of selected-root coordinates, and let

Thus \dim H_P=g. We regard H_P^* as the space of covectors on E_P annihilating all loop responses.

The next lemma supplies the common subspace promised at the start of this section. Its formal-coordinate argument is the reason we obtain compatible relations between pairs of blocks.

We henceforth write \mathcal G for these compatible spaces, specifying the blocks when needed. We have identified all linear relations shared by the loop and paired corrections. The next section proves that the leading free-word errors satisfy the corresponding cyclic relations.

9 Cyclic relations on the leading free-word errors

The space \mathcal G controls the linear corrections available at several blocks. We must show that the errors to be corrected satisfy its relations for individual cyclic orbits of free words. Commuting scalar substitutions would combine too many words. We instead substitute matrices, repeat each block, and attach hubs whose derivative columns identify the possible left nullspace.

Traces of words and their cyclic symmetry are classical in matrix invariant theory; see [16, §4]. Here the response relations must first be proved to annihilate every matrix trace test. An explicit directed-cycle calculation then isolates each cyclic orbit.

Fix m\geq1 independent generic size-b points P_i. Write I_i for their selected root labels, so |I_i|=a, and let \mathcal G\subseteq\bigoplus_i H_{P_i}^* be the space of Proposition 8.5. As before, all scalars may be extended to a common field containing the required splitting data.

We prove the theorem in three steps. First, arbitrary matrix substitutions turn the free-word errors into matrices on a repeat factor. Second, a derivative test shows that every relation in \mathcal G annihilates their traces. Finally, directed-cycle matrices isolate each cyclic orbit.

Keep a choice as in Lemma 9.2. Adjoin L+1 independent generic size-b hub blocks, indexed by p=1,\ldots,L and by *, with spectra disjoint from the old blocks. The individual hub p sends an input arrow to repeat position p of every old block i, with fixed generic value U_{i,p}. The common hub * sends the same generic value U_{i,*} to every repeat position in old block i. There are no reverse arrows at this test input. Arrow values for different pairs of old blocks and hubs can be chosen independently and to satisfy the surjectivity conditions of Lemma 8.3.

Figure 2 shows the distinction between the individual hubs and the common hub.

Treat the entire deformed leaf array as one block and order it before the hubs. Over the \delta-Laurent field this is a star with maximal root count. Put

Select the L branches over each old selected label and the selected labels of all hubs, for a total of Na output rows. The rank of their derivative matrix is at most \rho_{bN}\leq Na-g. This bound follows from the original trace-minor identities, evaluated coefficientwise at the present Taylor input; it does not require the specialized input to be generic.

We now use the individual hubs to remove off-diagonal repeat-factor weights and the common hub to make all diagonal weights agree. The rank bound will then show that no element of \mathcal G has been lost.

The leading errors now satisfy every cyclic relation imposed by \mathcal G. The colored correction graph will make these exactly the annihilators of its order-by-order correction map.

10 Solving the free-series equations

We now use the response and cyclic-sum results to keep the output of a substitution conjugate to its constant term. The input retains two prescribed linear coefficients. This is the point at which free words, rather than commuting parameters, enter the construction essentially.

Work over a common split scalar field E in which the preceding evaluations and spectral projections are defined. All series substitutions below have small constant terms and are interpreted by the coefficientwise valuation convergence established earlier. In particular, the letters of a free series commute with its matrix coefficients; the letters do not commute with each other.

Put m=11 and index the blocks by i\in\mathbb Z/m\mathbb Z. Let P_i be jointly independent small generic pairs of b\times b matrices. Set R=\operatorname{diag}(P_i), a pair of mb\times mb matrices. The old coefficient algebra acts on a separate factor of dimension d. Write \lambda_{i,h} and \pi_{i,h} for the distinct eigenvalues and spectral projections of w(P_i). Eigenvalues belonging to different blocks are distinct. For a substitution Z with constant term R, let S_{i,h}(Z) denote the fixed-root Schur expression at \lambda_{i,h}, using the constant projection \pi_{i,h} in the i-th block.

We recall precisely the response information needed here. In each block choose one eigenvalue label from every part of the active partition, giving a selected labels. The selected loop-response map is

Put L_i=\operatorname{im}D_i and H_i=E^a/L_i. Proposition 8.5 supplies a vector space G=\mathcal G of dimension g and coordinate isomorphisms

For a generic fixed input arrow U on i\leftarrow j, varying its opposite arrow V gives two selected responses

The annihilator of these paired responses in H_i^*\oplus H_j^* is exactly

The identifications p_i are compatible for the whole collection of blocks and for all the generic arrows used below.

The transfer result supplies the following additional facts. If all Schur expressions vanish in degrees less than \ell, their degree-\ell coefficients are scalar on the projected eigenspaces. The resulting tuples belong to their active spaces and are determined by their selected entries. Denote the selected error for a word \omega of length \ell in block i by e_{i,\omega}\in E^a. The cyclic-sum result says that, for every orbit \mathcal O of words under cyclic rotation,

These statements hold for every partial substitution under consideration, including after an extension of its scalar field. The preceding results therefore supply all the hypotheses used in the next construction.

Figure 3 separates the generating cycle from the full local arrow pattern. The role of the four offsets is now explicit. Avoiding returns of length one or two starts the induction without changing the linear data. Consecutive offsets within each color make its only obstructions independent of the vertex. The remaining obstructions are exactly the cyclic sums already proved to vanish.

10.1 Fixed roots and Schur complements

The following elementary fact identifies the spectral equations with a single polynomial identity. Its coefficient ring need not be commutative. Let E be a field, let \mathfrak a=(s_1,s_2) be the augmentation ideal of the free algebra E\langle s_1,s_2\rangle, and put

Elements of E are central in T_N. Reduction modulo \mathfrak a is called the constant term.

For a matrix over E\langle\!\langle s_1,s_2\rangle\!\rangle, apply Lemma 10.2 after reduction to T_N. It follows that the fixed-root Schur equations hold through degree N exactly when the identity q(W)=0 holds through degree N. In particular, this equivalence applies at every finite stage of a formal correction, with the same polynomial q throughout.

11 Realization in a central division algebra

The preceding construction solved equations in split matrices. We now place its constant blocks and prescribed arrows in one central division algebra. Each subsequent correction is the solution of a finite linear system over its center. This permits descent without any assertion that solutions of nonlinear equations descend.

Throughout this section k has characteristic zero and contains all roots of unity, and A is a central division algebra of degree d over k. The integer b is the block size chosen earlier in the spectral case. In the case where q_0(w)=0 already holds as a free-series identity, any positive integer b may be used. Put m=11.

11.1 The two division-algebra factors

Choose independent central indeterminates \alpha,\beta and put k_0=k(\alpha,\beta). Let B_b be the symbol algebra over k_0 generated by u,v with

where \zeta_b is a primitive b-th root of unity. For b=1 this means B_1=k_0. By the construction in Section 4.2, B_b is central simple of degree b. Lemma 4.3 shows that A\otimes_k B_b is division and remains division under pure rational and central Laurent scalar extensions. We will use these facts twice in adjoining the cyclic block factor.

Fix a k_0-basis (e_a)_{a=1}^{b^2} of B_b. Over k_0 introduce the following jointly independent variables:

Let L_0 be their rational function field and let \sigma increase the index j by one, fixing k_0. This automorphism has order exactly m. Put H_0=L_0^{\langle\sigma\rangle}. Artin’s fixed-field theorem for a finite group of automorphisms gives a cyclic Galois extension L_0/H_0 of degree m; see [15, Corollary 3.11, p. 38].

Choose a further independent central indeterminate a_* and form the cyclic algebra C over F_0=H_0(a_*) with coefficient field L_0(a_*) and generator \tau, subject to

It has vector-space decomposition C=\bigoplus_{r=0}^{m-1}L_0(a_*)\tau^r. After extending scalars to a field containing L_0(a_*), represent c\in L_0 by the diagonal with entries \sigma^{-i}(c) and \tau by the cyclic forward shift, whose wrap entry is a_*. These matrices satisfy (11.1) and span all matrix units. This gives the usual splitting of C and shows that it is central simple of degree m; see [4, Construction 2.5.1 and Proposition 2.5.2]. Our conjugation convention uses the inverse of their Galois generator.

Define

Then D_0 is central simple of degree dbm over F_0 by the tensor-product and scalar-extension properties of central simple algebras [14, Chapter IV, Corollary 2.8 and Proposition 2.15]. It is also division. Indeed

is division by the rational-extension argument. Extend \sigma to D_L by fixing the old algebra factors. The skew Laurent division ring D_L((t;\sigma)) receives a unital homomorphism from D_0 by \tau\mapsto t and a_*\mapsto t^m. The central element t^m is transcendental over H_0. Simplicity again makes this map injective, and finite dimension makes D_0 division.

Finally put

The central Laurent argument shows that D is a central division algebra over F of degree dbm. Input coefficients will lie in \widetilde B, so that they commute with the old factor A.

11.2 One realization of all generic data

For r\in\mathbb Z/m\mathbb Z, let

This notation uses the evident coefficient embedding and scalar extension; equivalently it is the F-span of that graded summand in \widetilde B_0. Upon splitting, \mathcal V_r consists exactly of all b\times b entries at offset r in the m-block matrix. In fact L_0\otimes_{H_0}E\simeq E^m over a splitting field E, and B_b becomes M_b(E). Multiplication by \tau^r moves these independent diagonal entries to offset r, with nonzero wrap factors. In particular \mathcal V_0 gives all diagonal blocks, and \mathcal V_{-r} gives all reverse entries for offset r.

Define the two constant inputs and four arrow pairs by

The prescribed initial substitution is the pair

After splitting, the diagonal blocks of (P_1^0,P_2^0) are exactly the required independent small generic pairs. The two input-component entries at every forward edge are independent generic entries, jointly with all the diagonal data.

Here independence is over the old split coefficient field, which is the field used in the generic rank tests. To check it, first split A and B_b using constants independent of the variables x,y. The m conjugates of each coefficient use the disjoint sets of variables with different index j. The split symbol basis is a matrix basis. Permuting those variables, changing that basis, and multiplying entries by powers of a_* preserve their algebraic independence. This verifies all generic determinant conditions simultaneously. No independence over the larger invariant field H_0 is asserted or needed.

We may perform the spectral calculations after extending F to a split Laurent field and then to the finite extensions needed for eigenvalues. Its constant coefficient field contains the old algebraic closure and the rational variables just chosen. The matrix constant terms remain \varepsilon-small. Arrow coefficients need not be small, since they occur only with positive free-word degree.

11.3 Linear descent of the correction

Assume first that q_0(w)\ne0. Let W_0=w(P_1^0,P_2^0)\in D and choose its monic minimal polynomial

We keep this polynomial fixed throughout the correction. It is irreducible, because F[W_0] is a finite-dimensional commutative domain, and separable because the characteristic is zero. In a splitting field its roots are exactly the distinct eigenvalues of the constant output. For completeness, left multiplication by W_0 on D has characteristic polynomial a power of q, since D is a vector space over F[W_0]. After splitting, that left multiplication is a direct sum of copies of the action of the split matrix W_0. Every root of q therefore occurs in that matrix, and no other eigenvalue occurs. Separability makes its action semisimple.

We solve q(w(Z))=0 in free series over D. Initially take (11.2). Its split Schur equations vanish through degree two, so Lemma 10.2 gives the same assertion for the polynomial relation. Suppose inductively that a partial substitution solves this relation through degree \ell-1, where \ell\ge3. Allow a new coefficient in \mathcal V_{-r}^2 at every word of length \ell-1, for r=1,2,3,4, and a new coefficient in \mathcal V_0^2 at every word of length \ell. These are finite-dimensional F-spaces.

Impose every coefficient equation of

This is a finite affine linear system over F in the allowed new coordinates. Indeed two new insertions have degree at least 2(\ell-1)>\ell. At any fixed free-word degree, only boundedly many positive-degree insertions occur in the evaluation of the old series; the remaining insertions are from the \varepsilon-small constant input. The finitely many previously fixed coefficients have a finite lower valuation bound. Thus the sums defining each coefficient of (11.4), and its linear part in the new coordinates, converge in D and are defined over F. Choosing an F-basis of D turns them into finitely many ordinary scalar linear equations.

After scalar extension, each permitted input space becomes exactly the collection of reverse edges or loops used in Proposition 10.1. The current partial substitution is a valid partial split solution by Lemma 10.2. The proposition supplies a split correction, and that lemma converts its Schur vanishings back to all the equations in (11.4). In particular the new degree-\ell-1 reverse coefficients preserve the already solved equations: their diagonal Schur variation at that degree is zero. We have therefore solved the whole finite linear system after a field extension. A matrix and its augmented matrix have the same ranks before and after extension, so the system already has a solution over F.

Choose such a solution at every step. The coefficient stabilization in Proposition 10.1 produces

with the prescribed constant and linear terms. Only finite linear systems have been descended. Neither the roots of q nor the split correction coefficients need belong to F.

Since the input coefficients commute with A, evaluation at Z defines a substitution homomorphism

It is well-defined by the same coefficientwise valuation bound: at fixed output degree, only boundedly many nonconstant input factors occur. The bound also justifies multiplication and regrouping of the convergent coefficient sums. Hence every old polynomial identity on the tuple and the chosen series is preserved by this substitution.

If q_0(w)=0 already holds, use the very same division algebra and the initial substitution (11.2), with any b\ge1. No spectral correction is required; the old relation q_0(w) remains zero under the substitution homomorphism. Both cases produce a central division algebra of degree 11bd>d, with identical generic forms for the constant and the two linear input coefficients. The next step restores generation and two transverse directions using these coefficients. That step is needed in the unconstrained case as well as in the spectral case.

12 Restoring generation and two transverse directions

The preceding construction gives a substitution Z(s_1,s_2) inside the centralizer of A in a larger central division algebra. It preserves the old polynomial identities and algebraizes the chosen series. To complete Theorem 2.1, we must make the new constant tuple generate the larger algebra and retain two independent directions modulo inner derivations. We first use the original transverse directions to prove generation of the generic diagonal product and to control the arrows between its blocks. Translating the first free letter by a new central parameter then joins the blocks into a full matrix algebra. A single rank calculation shows that both directions remain independent modulo inner derivations.

Throughout this section, brackets with a tuple are taken componentwise. Thus, for C=(C_1,\ldots,C_u) in an algebra B, write

Polynomial relations on X_0 detect the two input directions to first order.

In particular, there are f_1,f_2\in I whose derivative pairs are (1,0) and (0,1). These relations recover the two generic input matrices from small evaluations of the tuple.

All determinant conditions in Lemma 12.2 hold on the same independent block and arrow coordinates used in the division realization. Only finitely many blocks and arrows are involved, so their nonzero minors can be imposed simultaneously. In particular, the argument does not require replacing that realization by a different choice of data.

We next isolate the translation argument. Its hypotheses concern only the constant and linear coefficients; higher-order corrections may be arbitrary.

Apply the lemma to Y=X(Z). Lemma 12.2 supplies its diagonal-product and edge hypotheses, while the eleven-block construction supplies precisely the two color supports. Consequently the translated tuple generates after splitting and has two independent tangent classes. These are finite rank conditions, so they also hold over the division algebra’s own center. The central Laurent extension by \gamma preserves division. Both the old polynomial identities and the new algebraic relation are preserved by the convergent substitution.

12.1 Returning to a countable center

The complete Laurent fields used so far need not be countable. We finish by retaining only the scalar data required by the result. Use the notation of Section 11: F_0=H_0(a_*) is the construction field, D_0 is the central division algebra over it, and

is the full shifted Laurent center. The field F_0 is countable: L_0 is generated by finitely many indeterminates over the countable field k(\alpha,\beta), and H_0 is a subfield of L_0. The algebra D_0 has degree dbm, and D_0\otimes_{F_0}L is the central division algebra just constructed.

Choose an F_0-basis of D_0. Let k'\subseteq L be the field generated by F_0, the coefficients of the monic polynomial q (taken to be q_0 when q_0(w)=0), and every scalar coordinate in this basis of every coefficient of the translated series X' and w'. There are countably many free words and finitely many coordinates per coefficient, so k' is countable. Put

This is central simple of degree dbm. Its scalar extension to L is division, so A' embeds in a division ring and has no zero divisors. A finite-dimensional algebra without zero divisors is a division algebra. Thus A'/k' has the required central division property and strictly larger degree.

All coefficients of X',w' and q lie in this smaller model. The identities descend by its injective scalar extension to L. Generation and transverse independence descend as well, since the ranks of the finite coordinate matrices defining them do not change under field extension. This proves every conclusion of Theorem 2.1.

In the unconstrained case q_0(w)=0, the same argument applies without the spectral corrections: use the same eleven-block generic constant and colored linear terms, retain the relation q_0, and carry out the translation and countable restriction above. Thus that case also increases degree while preserving generation and both directions.