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 (Kurosch 1941, 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 (Golod 1964, 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 (Jacobson 1954, 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 (Bell et al. 2013, 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 (Amitsur 1956, 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 (Goodearl and Letzter 2026) treats nonfinite generation as an algebra for classes of transcendental skew fields and explicitly distinguishes that question from the algebraic Kurosh problem. Theorem 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 (Gille and Szamuely 2006, sec. 2.5). Traces organize cyclic word relations (Procesi 1976, sec. 4), while inner derivations describe infinitesimal conjugation (Jacobson 1937, 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.

The construction

The main ingredient is Theorem 2. 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. 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.

Scheduling all polynomial expressions

Start with \[k_0=\overline{\mathbb Q},\qquad A_0=k_0,\qquad X^{(0)}=(z_1,z_2).\] 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.

Inductively we obtain countable fields \[k_0\subseteq k_1\subseteq k_2\subseteq\cdots,\] finite-dimensional central division algebras \(A_i/k_i\) of strictly increasing degrees, and pairs \[X^{(i)}\in A_i\langle\!\langle z_1,z_2\rangle\!\rangle^2\] satisfying the same generation and transverse-independence hypotheses. Write \((x_i,y_i)=X^{(i)}_0\) for the constant pair; in particular, \[ A_i=k_i[x_i,y_i]. \tag{2}\]

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 \(n\)th pair, beginning at \(n=0\), then \(i_n\leq n\). At stage \(n\), apply the lifting theorem to \[w=p_{i_n,j_n}(X^{(n)}).\] 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 \[q_{i_n,j_n}(t)\in k_{n+1}[t]\] such that \[ q_{i_n,j_n}\bigl(p_{i_n,j_n}(X^{(n+1)})\bigr)=0. \tag{3}\] 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.

A two-generated algebraic division ring

Fix a nonprincipal ultrafilter \(\mathcal U\) on \(\mathbb N\) and form (see (Keisler 2010) for the general construction) \[\mathcal D=\prod_{i\geq0} A_i\big/\mathcal U.\] 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 \[k_\infty\hookrightarrow\mathcal D:\] 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 \[x=[(x_i)_i],\qquad y=[(y_i)_i],\qquad R=k_\infty[x,y]\subseteq\mathcal D.\] Every \(r\in R\) has the form \(p(x,y)\) for some \(p\in k_j\langle x,y\rangle\). The schedule and (3) 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 \[a_0+a_1r+\cdots+a_nr^n=0, \qquad a_j\in k_\infty.\] Its constant coefficient is nonzero: otherwise multiplication by \(r^{-1}\) in the ambient division ring \(\mathcal D\) would give a shorter relation. Consequently \[r^{-1}=-a_0^{-1}(a_1+a_2r+\cdots+a_nr^{n-1})\in R.\] Since \(k_\infty\) is countable and \(R\) is generated by two elements as an algebra, \(R\) is countable. Its characteristic is zero.

Infinite dimension and passage to the center

We first prove \[ [R:k_\infty]=\infty. \tag{4}\] 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 (2), 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 \[\dim_{k_i}V_{i,\ell}\geq\ell+1\] 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 \[\sum_{j=0}^{\ell} c_j w_j(x,y)=0\] 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 (4).

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 \[R=k_\infty[x,y]=F[x,y]=F(x,y),\] 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 (Deo et al. 2019, sec. 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 \[b_i b_j=\sum_{h=1}^n c_{ijh}b_h, \qquad x=\sum_{h=1}^n u_hb_h, \qquad y=\sum_{h=1}^n v_hb_h,\] 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 \[B=\sum_{h=1}^n E b_h\] 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 \[[R:k_\infty]\leq n[E:k_\infty]<\infty,\] contrary to (4).

Thus the countable characteristic-zero division ring \(D=R\) is algebraic over its actual center \(F\), while \[D=F[x,y]=F(x,y),\qquad [D:F]=\infty.\] The finite set \(\{x,y\}\) witnesses the failure of local finiteness.

This proves Theorem 1, subject only to Theorem 2, which we now prove.

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 (Milne 2020a, 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 3, 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 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 \[K[[P]]\longrightarrow C(P^\circ)((\varepsilon)), \qquad P\longmapsto\varepsilon P^\circ,\] is injective, by homogeneous degree or by Lemma 4. 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 \[ K[[P]]\longrightarrow E[[\Delta]],\qquad P\longmapsto R+\Delta \tag{5}\] 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.

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 \(n\)th root of unity \(\zeta_n\in k\). Over \(k_0=k(\alpha,\beta)\) let \(B_n\) be generated by \(x,y\) with \[x^n=\alpha,\qquad y^n=\beta,\qquad yx=\zeta_nxy.\] 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 \(n\)th 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 (Gille and Szamuely 2006, 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 \[P_i=\varepsilon\sum_j c_{ij}b_j.\] This gives an element of a central division algebra of degree \(dn\) over \[F_n=k(\alpha,\beta,(c_{ij}))((\varepsilon)).\] 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 \[ t_n=\frac{dn}{e_n}. \tag{6}\] 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 \(N\)th 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 \(\operatorname{Frac}K[[P]]\).

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 \[t=t_p,\qquad \mathcal S=\{pN:N\ge1\}.\]

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 (5); a labeling is chosen by reduction to the roots at the base.

Let \(\rho_n\) be the generic rank of the input Jacobian of \[\operatorname{tr}W,\operatorname{tr}W^2,\ldots, \operatorname{tr}W^{dn}.\] Its minors are formal entry identities, so \(\rho_n\) bounds the corresponding specialized ranks. At a maximal-root-count point the scalar branches satisfy \[d\operatorname{tr}W^j =jt\sum_{h=1}^{e_n}\lambda_h^{j-1}\,d\lambda_h.\] 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.

Loop and paired responses

Upper triangular block evaluations are also the basic mechanism of noncommutative difference-differential calculus; see (Kaliuzhnyi-Verbovetskyi and Vinnikov 2014, 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 \[C_h(\mathcal W)=\bar\pi(\mathcal W-\lambda_h)\bar\pi.\] 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 \[ S_h(\mathcal W)= \pi(\mathcal W-\lambda_h)\pi -\pi\mathcal W\bar\pi C_h(\mathcal W)^{-1} \bar\pi\mathcal W\pi. \tag{7}\] This expression records the obstruction to keeping the root \(\lambda_h\) fixed. For \(u\) in the image of \(\pi\), the vector \[u-C_h(\mathcal W)^{-1}\bar\pi\mathcal W\pi u\] 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 27 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 \[ \pi_h W'_P(Y)\pi_h=D_h(P;Y)\pi_h. \tag{8}\] This compression is scalar: apply Lemma 8 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 6, each of these linear functionals is nonzero. Because \(K\) is infinite, we can choose a constant direction \(v\) over \(K\) with \[ D_h(P;v)\ne0\qquad(1\le h\le e_n). \tag{9}\] 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 \[T=\begin{pmatrix}P&U&0\\0&Q&V\\0&0&P\end{pmatrix}.\] Set \[ M_h(P,Q;U,V)=\pi_h\left( W(T)_{13} -W(T)_{12}(W(Q)-\lambda_h)^{-1}W(T)_{23} \right)\pi_h. \tag{10}\] Block paths show that this is bilinear in \(U,V\). The inverse exists because the spectra at independent points are disjoint.

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 (10) has a formal expansion in the entries of \(Q\) over \(\Omega_P\). Define \[m_h=\frac1t\operatorname{tr}M_h, \qquad m=(m_h)_{h=1}^{e_n},\qquad D=(D_h)_{h=1}^{e_n}.\] 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 9 holds also for these formal expansions, by clearing their determinants and applying Lemma 4.

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 \[a_n=\dim_{\Omega_P}\mathcal T_P.\] 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 3. 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.

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.

A formal identity for linked branches

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 \(a_n\).

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.

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 \[T_N=E\langle s_1,s_2\rangle/\mathfrak a^{N+1} \qquad (N\geq 0).\] 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 27 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.

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 \[u^b=\alpha,\qquad v^b=\beta,\qquad vu=\zeta_buv,\] 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 5 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: \[x_{c,a,j},\qquad y_{r,c,a,j}, \qquad \begin{matrix} c\in\{1,2\},\quad 1\le a\le b^2,\\ j\in\mathbb Z/m\mathbb Z,\quad r\in\{1,2,3,4\}. \end{matrix}\] 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 (Milne 2022, 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 \[ \tau^m=a_*,\qquad \tau c\tau^{-1}=\sigma(c) \quad(c\in L_0). \tag{31}\] 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 (31) and span all matrix units. This gives the usual splitting of \(C\) and shows that it is central simple of degree \(m\); see (Gille and Szamuely 2006, Construction 2.5.1 and Proposition 2.5.2). Our conjugation convention uses the inverse of their Galois generator.

Define \[\widetilde B_0=(B_b\otimes_{k_0}F_0)\otimes_{F_0}C, \qquad D_0=A\otimes_k\widetilde B_0.\] Then \(D_0\) is central simple of degree \(dbm\) over \(F_0\) by the tensor-product and scalar-extension properties of central simple algebras (Milne 2020b, IV, Corollary 2.8 and Proposition 2.15). It is also division. Indeed \[D_L=(A\otimes_k B_b)\otimes_{k_0}L_0\] 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 \[F=F_0((\varepsilon)),\qquad \widetilde B=\widetilde B_0\otimes_{F_0}F,\qquad D=D_0\otimes_{F_0}F.\] 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 \(A\).

One realization of all generic data

For \(r\in\mathbb Z/m\mathbb Z\), let \[\mathcal V_r= \bigl(B_b\otimes_{k_0}L_0\bigr)\tau^r \otimes_{H_0}F \ \subseteq\ \widetilde B.\] 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 \[P_c^0=\varepsilon\sum_a x_{c,a,0}e_a, \qquad U_{r,c}=\left(\sum_a y_{r,c,a,0}e_a\right)\tau^r \quad(c=1,2).\] The prescribed initial substitution is the pair \[ Z_c^{\mathrm{init}}(s_1,s_2) =P_c^0+(U_{1,c}+U_{2,c})s_1 +(U_{3,c}+U_{4,c})s_2. \tag{32}\] 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.

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 \[ q(t)\in F[t]. \tag{33}\] 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 (32). Its split Schur equations vanish through degree two, so Lemma 27 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 \[ q(w(Z))=0\pmod{(s_1,s_2)^{\ell+1}}. \tag{34}\] 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 (34), 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 26. The current partial substitution is a valid partial split solution by Lemma 27. The proposition supplies a split correction, and that lemma converts its Schur vanishings back to all the equations in (34). 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 26 produces \[Z\in\widetilde B\langle\!\langle s_1,s_2\rangle\!\rangle^2, \qquad q(w(Z))=0,\] 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 \[A\langle\!\langle z_1,z_2\rangle\!\rangle \longrightarrow D\langle\!\langle s_1,s_2\rangle\!\rangle.\] 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 (32), 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.

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 \[L=F_0((\varepsilon))((\gamma))\] 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 \[A'=D_0\otimes_{F_0} k'.\] 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.

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.