A Factorization Theory for some Free Fields

Although in general there is no meaningful concept of factorization in fields, that in free associative algebras (over a commutative field) can be extended to their respective free field (universal field of fractions) on the level of minimal linear representations. We establish a factorization theory by providing an alternative definition of left (and right) divisibility based on the rank of an element and show that it coincides with the classical left (and right) divisibility for non-commutative polynomials. Additionally we present an approach to factorize elements, in particular rational formal power series, into their (generalized) atoms. The problem is reduced to solving a system of polynomial equations with commuting unknowns.


Introduction
From an algebraic point of view fields are usually not very interesting due to the lack of "structure", for example, they do not have non-zero non-units. However here, the field -the universal field of fractions ("free field") of the free associative algebra (over a commutative ground field)-is non-commutative and infinite dimensional over its center (at least if we exclude the one-variable case). A brief introduction can be found in [Coh03b,Secton 9.3], for details we refer to [Coh85,Chapter 7] and [Coh06, Chapter 7], where also historical information is provided: "Until 1970 the only purely algebraic methods of embedding rings in fields were based on Ore's method [Ore31]." The main idea is to view all elements in terms of their normal form (minimal linear representation) [CR94]. Given an element, the dimension of a minimal linear representation defines its rank [CR99], for example, the rank of a word/monomial of length n is n + 1. Since multiplication (of two elements) can be formulated in terms of linear representations, we establish a concept to "reverse" this step, that is, given an element (by a minimal linear representation) to find left (and right) divisors subject to conditions of the ranks of the involved elements.
In [Sch17b] we showed that in the free associative algebra there is a rather natural correspondence between a factorization of an element and (upper right) blocks of zeros in (a special form of) its minimal linear representations. One does not have to take care about the ranks. In general a minimal multiplication, that is, a multiplication on the level of minimal linear representations is much more subtle. Now, how do we have to define divisibility in terms of the rank such that it is equivalent to that in the free associative algebra?
Joining factorization theory in the non-commutative setting -for an overview see [Sme15]-and the theory of embedding "non-commutative" rings into a (skew) field -to be more precise: embedding free ideal rings (firs) into their respective universal field of fractions, see [Coh06, Chapter 2]-even for the "simplest" case of the free associative algebra results in a very rich structure, maybe not only for a "free factorization theory". Somewhat paradoxical is the fact that the inverse plays a crucial role. Since each non-zero element (in the free field) is invertible, we can use both, its rank and that of its inverse, for example, the inverse of a polynomial of rank n ≥ 2 has rank n − 1. A corollary to the minimal inverse (Theorem 2.18) is used to identify trivial units, that is, units from the (commutative) ground field. We do not even have to exclude the one-variable case. After fixing the basic notation and stating the basic definitions in Section 1, we develop the main (technical) tools in Section 2. In a first reading only Proposition 2.1 (rational operations) and Theorem 2.18 (minimal inverse) are important. The main part is Section 3 where the factorization theory is developed, starting with Definition 3.1 and culminating (but not ending) in Theorem 3.10. Finally, in Section 4, minimal multiplication (Theorem 4.2) and factorization (Theorem 4.8) is discussed.
Remark. This exposition is not meant to serve as an introduction, neither to free fields nor to non-commutative factorization (in free associative algebras). Instead, depending on the background, the example in [Sch17b,Section 4], the connection to formal power series [Sch17a,Section 3] or the polynomial factorization [Sch17b, Section 2] might be helpful. One way to get acquainted with free fields is to use it ("almost" like the rational numbers) and explore the rich theory in parallel. The step from inverting a non-zero number, say in s = v a or as = v with unique solution s, to inverting "full" matrices is non-trivial but similar: As = v with unique solution vector s (we are usually interested in its first component s 1 ).

Preliminaries
Definition 1.1 (Inner Rank, Full Matrix, Hollow Matrix [Coh85], [CR99]). Given a matrix A ∈ K X n×n , the inner rank of A is the smallest number m ∈ N such that there exists a factorization A = T U with T ∈ K X n×m and U ∈ K X m×n . The matrix A is called full if m = n, non-full otherwise. It is called hollow if it contains a zero submatrix of size k × l with k + l > n.
Definition 1.2 (Associated and Stably Associated Matrices [Coh95]). Two matrices A and B over K X (of the same size) are called associated over a subring R ⊆ K X if there exist invertible matrices P, Q over R such that A = P BQ. A and B (not necessarily of the same size) are called stably associated if A ⊕ I p and B ⊕ I q are associated for some unit matrices I p and I q . Here by C ⊕ D we denote the diagonal sum C .
. D . Lemma 1.3 ([Coh95, Corollary 6.3.6]). A linear square matrix over K X which is not full is associated over K to a linear hollow matrix.
Remark. A hollow square matrix cannot be full [Coh85,Section 3.2], illustrated in an example: Definition 1.4 (Linear Representations, Dimension, Rank [CR94], [CR99]) It is called minimal if A has the smallest possible dimension among all linear representations of f . The "empty" representation π = (, , ) is the minimal one of 0 ∈ F with dim π = 0. Let f ∈ F and π be a minimal linear representation of f . Then the rank of f is defined as rank f = dim π.
Remark. Cohn and Reutenauer define linear representations slightly more general, namely f = c + uA −1 v with possibly non-zero c ∈ K and call it pure when c = 0.
are equivalent (pure) linear representations, of which the first is minimal, then the second is isomorphic to a representation π = (u, A, v) which has the block decomposition Definition 1.6 (Left and Right Families [CR94]). Let π = (u, A, v) be a linear representation of f ∈ F of dimension n. The families (s 1 , s 2 , . . . , s n ) ⊆ F with s i = (A −1 v) i and (t 1 , t 2 , . . . , t n ) ⊆ F with t j = (uA −1 ) j are called left family and right family respectively. L(π) = span{s 1 , s 2 , . . . , s n } and R(π) = span{t 1 , t 2 , . . . , t n } denote their linear spans (over K).
Proposition 1.7 ([CR94, Proposition 4.7]). A representation π = (u, A, v) of an element f ∈ F is minimal if and only if both, the left family and the right family are K-linearly independent.
Definition 1.8 (Admissible Linear Systems [Coh72], Admissible Transformations [Sch17a]). A linear representation A = (u, A, v) of f ∈ F is called admissible linear system (ALS) for f , written also as As = v, if u = e 1 = [1, 0, . . . , 0]. The element f is then the first component of the (unique) solution vector s. Given a linear representation A = (u, A, v) of dimension n of f ∈ F and invertible matrices P, Q ∈ K n×n , the transformed P AQ = (uQ, P AQ, P v) is again a linear representation (of f ). If A is an ALS, the transformation (P, Q) is called admissible if the first row of Q is e 1 = [1, 0, . . . , 0].
Remark. The left family (A −1 v) i (respectively the right family (uA −1 ) j ) and the solution vector s of As = v (respectively t of u = tA) are used synonymously.
Transformations can be done by elementary row-and column operations, explained in detail in [Sch17a, Remark 1.12]. For further remarks and connections to the related concepts of linearization and realization see [Sch17a, Section 1].
For elements in the free associative algebra K X a special form (with an upper unitriangular system matrix) can be used. It plays a crucial role in the factorization of polynomials because it allows to formulate a minimal polynomial multiplication (Proposition 2.17) and upper unitriangular transformation matrices (invertible by definition) suffice to find all possible factors (up to trivial units). For details we refer to [Sch17b, Section 2].
Remark. The following definition of pre-standard admissible transformations is slightly more general than that in [Sch17b, Definition 2.1]. Definition 1.9 (Pre-Standard ALS, Pre-Standard Admissible Transformation). An ALS A = (u, A, v) of dimension n with system matrix A = (a ij ) for a non-zero polynomial 0 = p ∈ K X is called pre-standard , if (1) v = [0, . . . , 0, λ] ⊤ for some λ ∈ K and (2) a ii = 1 for i = 1, 2, . . . , n and a ij = 0 for i > j, that is, A is upper triangular.
A pre-standard ALS is also written as A = (1, A, λ) with 1, λ ∈ K. An admissible transformation (P, Q) for an ALS A is called pre-standard , if the transformed system P AQ is (again) pre-standard. q ∈ pH = {ph | h ∈ H}. Two elements p, q are called left coprime if for all h such that h | l p and h | l q implies h ∈ H × = {f ∈ H | f is invertible}, that is, h is an element of the group of units. Right division p | r q and the notion of right coprime is defined in a similar way. Two elements are called coprime if they are left and right coprime.
Definition 1.13 (Atomic Domains [BS15, Section 2]). Let R be a domain and H = R • . An element p ∈ H \ H × , that is, a non-zero non-unit (in R), is called an atom (or irreducible) if p = q 1 q 2 with q 1 , q 2 ∈ H implies that either q 1 ∈ H × or q 2 ∈ H × . The set of atoms in R is denoted by A(R). The (cancellative) monoid H is called atomic if every non-unit can be written as a finite product of atoms of H. The domain R is called atomic if the monoid R • is atomic.
Definition 1.14 (Similarity Unique Factorization Domains [Sme15, Definition 4.1]). A domain R is called similarity factorial (or a similarity-UFD ) if R is atomic and it satisfies the property that if p 1 p 2 · · · p m = q 1 q 2 · · · q n for atoms (irreducible elements) p i , q j ∈ R, then m = n and there exists a permutation σ ∈ S m such that p i is similar to q σ(i) for all i ∈ 1, 2, . . . , m.

Rational Operations
Usually we want to construct minimal admissible linear systems (out of minimal ones), that is, perform "minimal" rational operations. Minimal scalar multiplication is trivial. In some special cases minimal addition can be formulated (Proposition 2.3). For minimal multiplication we refer to Section 4. For the minimal inverse we have to distinguish four cases, which are summarized in Theorem 2.18.
Proposition 2.1 (Rational Operations [CR99]). Let 0 = f, g ∈ F be given by the admissible linear systems A f = (u f , A f , v f ) and A g = (u g , A g , v g ) respectively and let 0 = µ ∈ K. Then admissible linear systems for the rational operations can be obtained as follows: The scalar multiplication µf is given by The sum f + g is given by The product f g is given by And the inverse f −1 is given by Definition 2.2 (Disjoint Elements [CR99]). Two elements f, g ∈ F are called disjoint if rank(f + g) = rank(f ) + rank(g).
Remark. Two polynomials are never disjoint. This can be easily seen in the construction (of an ALS) for the sum (of the polynomials). See also [CR99, Theorem 2.3].
For disjoint elements the formulation of a minimal addition (Proposition 2.3) is immediate. Testing if two elements are disjoint in F is difficult because it relies on techniques for minimizing linear representations. This is considered in future work. For regular elements, that is, elements in K rat X , existing algorithms (for constructing minimal linear representations) like [CC80] or [FM80] can be used. See also [Sch17a, Section 3]. However, since minimality of a linear representation is equivalent to K-linear independence of its left and right family respectively (Proposition 1.7), two elements are disjoint if all the components of their left and right families are from linear independent classes of the free field, that is, the union of two K-linearly independent subsets of different classes is K-linearly independent, for example Proposition 2.3 (Minimal Disjoint Addition). Let f, g ∈ F be disjoint and given by the minimal admissible linear systems A f = (u f , A f , v f ) and A g = (u g , A g , v g ) of dimension n f and n g respectively. Then the system of dimension n f + n g (from Proposition 2.1) for f + g is minimal.
Like in the polynomial case, factorization and minimal multiplication are tight together as opposite points of view. Further assumptions that guarantee minimality are developed in Section 3. They eventually enter in Theorem 4.2. Since we need alternative constructions (to that in Proposition 2.1) for the product several times we state them already here in Propositions 2.8 and 2.11. These constructions are used in particular in Theorem 4.2. Before, we need some technical results from [Sch17a] and [Sch17b]. However these are rearranged such that similarities become more obvious and the flexibility in applications is increased. In particular we proof Lemma 2.6 by applying Lemma 2.4.
Remark. The following lemmas are just like [Sch17b, Lemma 2.3] including the trival case for n = 1, that is T = 0 in Lemma 2.4 and U = 0 in Lemma 2.5.
Proof. Without loss of generality, assume that v = [0, . . . , 0, 1] ⊤ and the left family s = A −1 v is (s 1 , s 2 , . . . , s n−1 , 1). Otherwise it can be brought to this form by some admissible transformation (P • , Q • ). Now letĀ denote the upper left (n − 1) × (n − 1) block of A, lets = (s 1 , . . . , s n−1 ) and write As = v as and apply Lemma 2.4 to get the matrix T = [T , τ ] ∈ K 1×n such that B = T A. Thus we get the transformation Remark. If g is of type ( * , 1) then, by Lemma 2.6, each minimal ALS for g can be transformed into one with a last row of the form [0, . . . , 0, 1]. If g is of type (1, * ) then, by Lemma 2.7, each minimal ALS for g can be transformed into one with a first column of the form [1, 0, . . . , 0] ⊤ . This can be done by linear techniques, see the remark before [Sch17a, Theorem 4.20].
Since p ∈ K X is of type (1, 1), both constructions can be used for the minimal polynomial multiplication (Proposition 2.17). One could call the multiplication from Proposition 2.1 type ( * , * ). A necessary condition for minimality however is, that the left factor is of type ( * , 0) and the right factor is of type (0, * ), thus we will use this construction later as type (0, 0). Section 3 is dedicated to a sufficient condition. See also Figure 1.
Proposition 2.8 (Multiplication Type (1, * )). Let f, g ∈ F \ K be given by the admissible linear systems and A g = (u g , A g , v g ) = (1, A g , λ g ) of dimension n g respectively. Then an ALS for f g of dimension n = n f + n g − 1 is given by of dimension n f + n g for the product f g using Proposition 2.1. Add λ f -times column n f to column (n f + 1) (in the system matrix A ′ ). Remove column n f from A ′ and v ′ and row n f from A ′ and u ′ to get the ALS (2.10) of dimension n f + n g − 1.
Proposition 2.11 (Multiplication Type ( * , 1)). Let f, g ∈ F \ K be given by the admissible linear systems respectively. Then an ALS for f g of dimension n = n f + n g − 1 is given by Remove row (n f + 1) from A ′ and v ′ and column (n f + 1) from A ′ and u ′ to get the ALS (2.13) of dimension n f + n g − 1.
Remark. Recall that, if f (respectively g) is given by a minimal ALS, it can be transformed appropriately by Lemma 2.6 (respectively Lemma 2.7) into the form (2.9) (respectively (2.12)).
Lemma 2.14 is a slightly more general version of [Sch17b, Lemma 2.4]. The proof itself does not have to be adapted. The proof of (the following) Proposition 2.17 becomes simple by the help of the two lemmas 2.15 and 2.16 which are extracted of the original proof (of the minimal polynomial multiplication). They are useful later, especially in Lemma 3.7.
Remark. Note that the transformation in the following lemma is not necessarily admissible. However, except for n = 2 (which can be treated by permuting the last two elements in the left family), it can be choosen such that it is admissible.
Lemma 2.15. Let p ∈ K X \ K and g ∈ F \ K be given by the minimal admissible linear systems A p = (u p , A p , v p ) and A g = (u g , A g , v g ) of dimension n p and n g respectively with 1 ∈ R(g). Then the left family of the ALS A = (u, A, v) for pg of dimension n = n p + n g − 1 from Proposition 2.11 is K-linearly independent.
of A is K-linearly independent. Assume to the contrary that there is an index 1 < m ≤ n p such that (s m−1 , s m , . . . , s n ) is K-linearly dependent while (s m , . . . , s n ) is Klinearly independent. Then, by Lemma 2.14, there exist matrices T, U ∈ K 1×(n−m+1) as blocks in (invertible) matrices P, Q ∈ K n×n , is not necessarily admissible. But this is not an issue here, since we are only checking linear independence of the left family.) LetP (respectivelyQ) be the upper left part of P (respectively Q) of size n g × n g . Then the equation Nothing can be said about minimality of A since the right family t = uA −1 could be K-linearly dependent. As an example take p = xy and g = y −1 + z. An ALS for pg = x + xyz constructed by Proposition 2.11 is Lemma 2.16. Let f ∈ F \ K and q ∈ K X \ K be given by the minimal admissible of dimension n f and n q respectively with 1 ∈ L(f ). Then the right family of the ALS A = (u, A, v) for f q of dimension n = n f + n q − 1 from Proposition 2.8 is K-linearly independent.
Proposition 2.17 (Minimal Polynomial Multiplication [Sch17b, Proposition 2.6]). Let 0 = p, q ∈ K X be given by the minimal pre-standard admissible linear systems A p = (1, A p , λ p ) and A q = (1, A q , λ q ) of dimension n p , n q ≥ 2 respectively. Then the ALS A from Proposition 2.8 for pq is minimal of dimension n = n p + n q − 1.
Proof. Excluding the trival case p ∈ K or q ∈ K, the left family of A is K-linearly independent by Lemma 2.15 and its right family is K-linearly independent by Lemma 2.16. Whence A is minimal (by Proposition 1.7) and by construction in pre-standard form.
Theorem 2.18 (Minimal Inverse [Sch17a,Theorem 4.20]). Let f ∈ F \ K be given by the minimal admissible linear system A = (u, A, v) of dimension n. Then a minimal ALS for f −1 is given in the following way: (2.21) (Recall that the permutation matrix Σ reverses the order of rows/columns.) Remark. This simple consequence of Theorem 2.18 makes it possible to distinguish between trivial units (non-zero scalar elements) and non-trivial units, that is, elements in F \ K. The main idea in the factorization theory in Section 3 is to allow only (the insertion of) trivial units (in factorizations). It is used explicitly in Lemma 3.7 and implicitly in Theorem 3.10.

Factorization Theory
To compensate the lack of non-zero non-units in F = K( X ), that is, we will view the elements in terms of their minimal linear representations. Recall that the dimension of a minimal one of f ∈ F defines the rank of f . Firstly, in Definition 3.1, we define factors based on the rank. Although this definition would suffice to define divisibility for polynomials, it is too rigid in general. Since this is far from obvious it is explained in detail in an example before Definition 3.4 (left and right divisibility). Secondly, some preparation is necessary to be able to exclude the insertion of non-trivial units. This is the essence of Lemma 3.7. Finally, Theorem 3.10 yields, as the main result, the equivalence of the "classical" divisibility (in free associative algebras) and the new one (for the free field) for polynomials.
For a factorization of a (non-zero) polynomial p = q 1 q 2 · · · q m into atoms q i we would like to have a factorization of its inverse p −1 = (q 1 q 2 · · · q m ) −1 = q −1 m · · · q −1 2 q −1 1 into atoms q −1 i . For two polynomials p, q we have -due to the minimal polynomial multiplication-rank(p) + rank(q) = rank(pq) + 1. Recalling Definition 1.12 we have p | l h if h = pq for some q ∈ K X . The minimal inverse type (1, 1) yields rank(q −1 ) + rank(p −1 ) = rank(q) − 1 + rank(p) − 1 = rank(q −1 p −1 ), or q −1 "left divides" h −1 for h = pq. See Proposition 2.17, Theorem 2.18 and Lemma 3.5. To avoid inserting non-trivial units from F \ K, we have to bound the sum of the ranks of the two factors: rank(px) + rank(x −1 q) = rank(pq) + 2. For two polynomials p and q the previous definition tells us that p (respectively q) is a left (respectively right) factor of pq. However, in general f is not a left factor of h = f g. As an example take f = (xyz) −1 and g = x. Then rank(f ) + rank(g) = rank(z −1 y −1 x −1 ) + rank(x) = 3 + 2 > rank(z −1 y −1 ) + 1.
While here it is easy to see that f −1 and g have a non-trivial left divisor in K X (in the sense of Definition 1.12), this can be much more delicate in general, illustrated in Example 3.2. This example will also show that the definition of outer factors is rather restrictive and not applicable directly. Later left and right divisors will be defined more generally in such a way that outer factors can be "split off" in at least one possible sequence (see Definition 3.4). Although we will see later that this is a generalization of the factorization in the free associative algebra, it is much more difficult to apply for two reasons: One has to test all possible "sequences" of factorizations to get the atoms (up to "similarity"). And the invertibility of the transformation matrices -to admissibly transform the ALS in such a way that the factors can be "extracted"-has to be ensured by including a condition for nonvanishing determinant. The latter might restrict practical applications to rank ≤ 6, similar to the test if a matrix is full [Jan17]. Section 4 provides further details. Experminents show that testing (ir)reducibility of polynomials using pre-standard form works practically for rank ≤ 14 in some cases up to rank ≤ 17 [Jan17].
Remark. Even in "simple" cases it is difficult to decide whether an element is irreducible or not based on regular expressions. As an example take f = 1 − xy and g = (1 − zy) −1 . Then f g is irreducible while gf is reducible. One has to look on their minimal linear representations. Minimal admissible linear systems for f g and gf are respectively. The principle, how to find the left divisor g of gf , is explained in detail in the following example Then it is immediate (after recalling the construction of an ALS for the product from Proposition 2.1) that f 3 is a right factor of f by duplicating s 3 , that is, inserting a "dummy" row (between row 2 and 3): Thus, if a minimal ALS A = (u, A, v) for f is not of the form (3.3), we need to find an admissible transformation (P, Q) such that the (transformed) system matrix P AQ has a lower left zero block of size 2 × 2 and an upper right zero block of size 2 × 1 and only the last component of the right hand side P v is non-zero, to detect the right factor f 3 . Similarly, subtracting row 3 from 1 and adding column 2 to 4 in (3.3) yields Compare with Figure 1, k = 2 in type ( * , 1). By duplicating t 2 , that is, inserting a "dummy" column (between column 2 and 3) one can see that f 1 = (xy) −1 is a left factor of f = (xy) −1 (1 − xz)(yz) −1 : However, f 1 (respectively f 2 ) is not a left (respectively right) factor of f 1 f 2 while y −1 is a left factor of f 1 f 2 and x −1 is a left factor of x −1 f 2 . We now take a closer look on that phenomenon. A minimal ALS for f ′ = f 1 f 2 is given by The reason is, that the rank does not increase (when f 2 is multiplied by x −1 and y −1 from the left) because 1 ∈ R(x −1 f 2 ): By adding column 2 to column 1 and switching the first two rows (this results in switching the first two columns in t) we get [1, 0, 0] ⊤ as the first column in the system matrix (for the existence of these transformations see Lemma 2.7): After multiplying x −1 f 2 from the left by y −1 , we have 1 ∈ R(f ′ ). Hence a further multiplication by (for example) z −1 from the left increases the rank: To summarize, there are essentially two different factorizations (on the level of outer factors in Definition 3.1) of f = (xy) −1 (1 − xz)(yz) −1 , namely Remark. We have not yet checked, if this phenomenon could appear already in the regular case. Factorization on the level of realizations is discussed in [KVV09] and [HKV17]. Now we come to the main definition which will generalize that of left and right divisors in the free associative algebra (Definition 1.12). To be able to show in Theorem 3.10 that these definitions are indeed equivalent on K X , some preparation is necessary.
Remarks. Note that (iii) and (v) hold in particular for polynomials. Further, recall from Example 3.2 that for (iv) to hold in the case of p, q ∈ K X it is necessary but not sufficient that p and q are left coprime.
What is rather simple in a concrete example, namely to verify that we cannot "insert" non-trivial units turns out to be very technical since we have to investigate the left and right families in detail. One subtle step is discussed in the following example.
In particular the right family t = (t 1 , t 2 , t 3 , t 4 ) is K-linearly independent. Therefore we cannot write t 3 as a linear combination (over K) of h, t 1 and t 2 . This is trivial. The following not: When we invert a polynomial q (in terms of the minimal inverse), say q = z, we get type (0, 0) q −1 . What can happen, if we multiply a polynomial p with an element of type (1, 1) "containing" a non-trivial polynomial inverse?
Proof. For a fixed non-scalar polynomial p we consider factorizations p = p 1 p 2 · · · p m into m atoms p j . For notational simplicity let p 0 = p m+1 = 1. Here p 1 , p 2 , . . . , p m always denote atoms. Let r = min rank(p i0 p i0+1 · · · p m f −1 ) | p = p 1 p 2 · · · p m and i 0 ∈ {1, 2, . . . , m} and i 0 and p 1 p 2 · · · p m such that this minimum is attained. By assumption h = p i0 p i0+1 · · · p m f −1 ∈ F \ K with rank(h) = r. Thus f = h −1 p i0 p i0+1 · · · p m with non-scalar h. According to Theorem 2.18 there are four cases: • r ≥ 2 and rank(h −1 ) = r − 1 for type (1, 1), • r ≥ 2 and rank(h −1 ) = r for type (1, 0), • r ≥ 2 and rank(h −1 ) = r for type (0, 1) and • r ≥ 1 and rank(h −1 ) = r + 1 for type (0, 0). Now fix an arbitrary factorization of p (into atoms q i ) and any 1 < ℓ ≤ m and let p ′ = q 1 q 2 · · · q ℓ−1 and p ′′ = q ℓ q ℓ+1 · · · q m with ranks n ′ and n ′′ respectively. We have to show that We proceed as follows: Depending on the four cases we construct -using Proposition 2.8 and Proposition 2.11-admissible linear systems for n ′ h and h −1 n ′′ respectively and find an upper bound for the rows/columns that can be removed (due to K-linear dependent entries in their left and right families). We start by assuming type (1, 1). For n ′ h we construct an ALS A ′ of dimension n 1 = n ′ + r − 1 with block decomposition (as linear representation according to Theorem 1.5) For h −1 n ′′ we construct A ′′ of dimension n 2 = n ′′ + r − 2 with block decomposition Let k ′ t (respectively k ′ s ) be the size of block A ′ 1,1 (respectively A ′ 3,3 ) in π ′ and k ′′ t (respectively k ′′ s ) be the size of block A ′′ 1,1 (respectively A ′′ 3,3 ) in π ′′ . Firstly, we write the left and the right family of h −1 in terms of their respective family of h: Let (s h 1 , s h 2 , . . . , s h r ) and (t h 1 , t h 2 , . . . , t h r ) be the left and right family respectively of some minimal ALS for h. Then s h −1 = (1, s h r−1 , . . . , s h 2 )h −1 and t h −1 = h −1 (t h r−1 , . . . , t h 2 , 1) are the families of a minimal ALS for h −1 constructed by Theorem 2.18. Recall that row/column n ′ was eliminated in a system of dimension n ′ + r to get A ′ and row/column r was eliminated in a system of dimension r − 1 + n ′′ to get A ′′ . Secondly, we take a closer look at the left families of A ′ and A ′′ . They are (without loss of generality) respectively. The first observation is that k ′ s = 0, that is, the left family of A ′ is K-linearly independent because 1 ∈ R(h) and Lemma 2.15. The first r − 1 and the last n ′′ − 1 components of s ′′ are K-linearly independent. At most r − 1 (linear combinations of) components in s ′′ can be eliminated. Hence we have k ′′ s ≤ r − 1. However, we claim that However, we claim that Lemma 3.8. Let q ∈ H = K X • and p ∈ H = F • . Then p | F l q implies q = ph with p, h ∈ H.
Proof. For some m ′ < m, let p = f 1 f 2 · · · f m ′ and q = pf m ′ +1 · · · f m and σ = (i 1 , i 2 , . . . , i m ) be a permutation such that f i k is an outer factor of f σ 1,k f σ 2,k · · · f σ k,k for all k ∈ {2, 3, . . . , m}. We have to show that q k = f σ 1,k f σ 2,k · · · f σ k,k ∈ H and f i k ∈ H for all k = 1, 2, . . . , m by induction on k from m to 2. For k = m we have q m = q ∈ H. Without loss of generality assume that f i k is a proper right factor of q k (nothing has to be shown for trivial factors), that is, By assumption q k is a polynomial and thus all factorizations into atoms have the same length, say ℓ k . We claim that f i k = κ g ℓ0 · · · g ℓ k for some factorization q k = g 1 g 2 · · · g ℓ k , some ℓ 0 ∈ {1, 2, . . . , ℓ k } and κ ∈ K. Assume the contrary and apply Lemma 3.7 with p = q k and f = f i k to get the contradiction Theorem 3.10. Let p, q ∈ H = K X • . Then p | l q (respectively p | r q) if and only if p | F l q (respectively p | F r q) in F. Proof. Let p | l q, that is q ∈ pH = {ph | h ∈ H}. We show that p is a left factor of q = ph. By Lemma 3.5 (i) we get rank(q) = rank(p) + rank(p −1 q) − 1 and by (ii) we get rank(q −1 ) + 1 = rank(p −1 ) + 1 + rank(q −1 p), thus p is a left factor of ph = q and therefore (m ′ = 1 and m = 2 in Definition 3.4) p | F l q (in F). Conversely, we have to show that q ∈ pH. But this follows directly from the assumption p | F l q and Lemma 3.8.
Notation. Since the left (respectively right) division in K X is the same as in F we can simplify notation and use f | l g instead of f | F l g (respectively f | r g instead of f | F r g) in the following.
Definition 3.11 (Atoms, Irreducible Elements). Let H = F • . An element f ∈ H \ K, that is, a non-trivial unit (in F), is called (generalized) atom (or irreducible) if f = g 1 g 2 with g 1 , g 2 ∈ H and g 1 | l f implies that either g 1 ∈ K × or g 2 ∈ K × . Like in Definition 1.13, the set of atoms in F is denoted by A(F).

Proposition 3.12. A polynomial is an atom if and only if it is a generalized atom.
Proof. We have to show that A(K X ) = A(F) ∩ K X . Recall from Lemma 3.8 that for a polynomial p we have g 1 | l p implies p = g 1 g 2 with polynomials g 1 and g 2 . Now both implications are immediate.
Notation. In the following we use "atom" as the general term and "polynomial atom" if we want to emphasize that the atom is an element in the free associative algebra.

Minimal Multiplication and Factorization
Before we describe the correspondence of zero (lower left and upper right) blocks in the system matrix of a minimal ALS and a non-trivial factorization, we describe a construction of a minimal ALS A = (u, A, v) = (1, A, λ), A = (a ij ), for the product of two non-zero elements f, g given by minimal admissible linear systems, say of dimension n f and n g respectively, if f is a left factor of f g. According to Theorem 4.2 there are three cases (see also Figure 1): type lower left zeros "coupling" upper right zeros (1, * ) Note that for type (1, * ) and ( * , 1) the "coupling condition" must hold for each (admissibly) transformed system because otherwise both types could be "derived" easily from type (0, 0). To "reverse" the multiplication we need to transform an ALS accordingly using transformations of the form with entries α ij , β ij ∈ K. To ensure invertability we need det P = 0 and det Q = 0.
Remark. The minimal polynomial multiplication (Proposition 2.17) can be formulated as a corollary to the following theorem. The difficulty of the proof of the former is hidden in the definition of outer factors, Definition 3.1. To test if f is a left factor of f g in general relies on techniques for minimization of linear representations. This is considered in future work. Note that, in principle, Theorem 1.5 could be used directly for minimization. However, the difficulty is to test all possible block decompositions with non-linear techniques.
type ( * , 1) type (0, 0) Figure 1: There are three types of factorization of an element h = f g with rank(h) = n, rank(f ) = k and rank(g) = n − k for type (0, 0) or rank(g) = n − k + 1 otherwise. These types correspond to that of the minimal multiplication. For type (1, * ) and ( * , 1) the coupling has to be non-scalar for all transformations yielding appropriate zero blocks.
Theorem 4.2 (Minimal Multiplication). Let f, g ∈ F \ K be given by the minimal admissible linear systems A f = (u f , A f , v f ) and A g = (u g , A g , v g ) of dimension n f and n g respectively. Let n = n f + n g . If f is a left factor of f g, then a minimal ALS for f g is given by Proof. Since f is a left factor of f g, we have rank(f ) + rank(g) ≤ rank(f g) + 1, thus and hence minimality of A if 1 ∈ R(g) or 1 ∈ L(f ), that is, the first two cases/types ( * , 1) and (1, * ). For the last case/type (0, 0) we distinguish four subcases. Recall from Theorem 2.18 that if h is of type (1, 0) or (0, 1) and rank(h) + 1 if h is of type (0, 0).
Note that a priori we cannot assume minimality of A for f g, since this is what we have to prove. Therefore we cannot use the minimal inverse on the left hand side because we only know that, for example, 1 ∈ L(g) implies 1 ∈ L(A) by construction. However, by the minimal inverse, we know -since g is of type (0, * )-that g −1 is of type (1, 1) or (0, 1) and f −1 is of type (1, 1) or (1, 0). Hence we can use one of the first two cases and get rank(f g) ≥ rank(f ) + rank(g).
Remark. Let A = (a ij ) be the system matrix of the ALS from Theorem 4.2. For type (1, * ) there exists an i ∈ {1, 2, . . . , n f − 1} such that a i,n f is non-scalar. For type ( * , 1) there exists an j ∈ {n f + 1, n f + 2, . . . , n} such that a n f ,j is non-scalar. And for type (0, 0) the entries a i,n f +1 are scalar for i ∈ {1, 2, . . . , n f }. We refer to that as coupling conditions. Note that there is no transformation of the form (4.1) respecting the zero blocks yielding a "scalar coupling" in type (1, * ) (respectively type ( * , 1)) because that would contradict minimality of A f (respectively A g ).

Lemma 4.3.
Let h, f, g ∈ F \ K be given by the minimal admissible linear systems n, n f and n g respectively such that f is a left factor of h = f g.
Type (1, * ): If f is of type ( * , 1) then there exists an admissible transformation (P, Q) of the form (4.1) such that P AQ = (a i,j ) has • a lower left block of zeros of size n g × (n f − 1), • an upper right block of zeros of size (n f − 1) × (n g − 1) and • there exists an i ∈ {1, 2, . . . , n f − 1} such that a i,n f is non-scalar Type ( * , 1): If g is of type (1, * ) then there exists an admissible transformation (P, Q) of the form (4.1) such that P AQ = (a i,j ) has • a lower left block of zeros of size (n g − 1) × n f , • an upper right block of zeros of size (n f − 1) × (n g − 1) and • there exists an j ∈ {n f + 1, n f + 2, . . . , n} such that a n f ,j is non-scalar.
Type (0, 0): If f is of type ( * , 0) and g is of type (0, * ) then there exists an admissible transformation (P, Q) of the form (4.1) such that P AQ = (a i,j ) has • a lower left block of zeros of size n g × n f , • an upper right block of zeros of size n f × (n g − 1) and • a i,n f +1 ∈ K for i ∈ {1, 2, . . . , n f }.
Proof. Let A ′ = (u ′ , A ′ , v ′ ) = (1, A ′ , λ ′ ) be the minimal ALS for h = f g constructed by Theorem 4.2 from A f and λ λg A g . The system matrix A ′ = (a ′ ij ) has -by construction-appropriate (lower left and upper right) blocks of zeros andfor type (0, 0)-scalar entries a ′ i,n f +1 for i ∈ {1, 2, . . . , n f }. Since both systems A and A ′ for h are minimal, there exists, by Theorem 1.5, an admissible transformation (P, Q) such that P AQ = A ′ . The right hand side P v = v ′ does not change, hence (P, Q) is of the form (4.1). The coupling conditions are fullfilled due to the construction of the minimal multiplication.
Example 4.4. Let h = x −1 zy −1 x −1 be given by the minimal ALS Multiplication of type (0, 1) for n f = n g = 2 would violate the coupling condition, "creating" a non-minimal ALS for g in h = f g.
Lemma 4.5 (Factorization Type (1,  * )). Let h = f g ∈ F \ K be given by the minimal admissible linear system A = (u, A, v) = (1, A, λ) of dimension n ≥ 2 and fix 1 < k ≤ n. Assume that A has a lower left block of zeros of size (n − k + 1) × (k − 1) and an upper right block of zeros of size (k − 1) × (n − k). For a transformation (P, Q) let a ′ ij denote the entries of P AQ. If for each transformation (P, Q) of the form (4.1) respecting these zero blocks there exists an i ∈ {1, 2, . . . , k − 1} such that a ′ i,k is non-scalar then f is a left factor of type ( * , 1) of h with rank(f ) = k and rank(g) = n − k + 1.

Proof. By assumption, A is of the (block) form
with square diagonal blocks A 1,1 , A 2,2 and A 3,3 of size k−1, 1 and n−k−1 respectively. We duplicate the entry s k in the left family by inserting a "dummy" row (and column) to get the following ALS of size n + 1: that is, "reversing" the construction from Proposition 2.8. The subsystems of dimension k and n − k + 1 are minimal for f (due to the coupling condition) and g = µs k respectively, otherwise we could construct an ALS for h of dimension n ′ < n, contradicting minimality of A. Clearly, 1 ∈ L(f ). By construction we have rank(f ) + rank(g) = rank(h) + 1, thus we only have to show that rank(g −1 ) + rank(f −1 ) ≤ rank(h −1 ) + 1 for f to be a left factor of h by distinguishing four cases (like in the minimal multiplication) and apply the minimal inverse. If h is of type (1, 1), then f is of type (1, 1) and g is of type ( * , 1). Thus rank(g −1 ) ≤ rank(g) and we get rank(g −1 )+rank(f −1 ) ≤ rank(g) + rank(f ) − 1 = rank(h −1 ) + 1. The other cases are as easy.
Lemma 4.6 (Factorization Type ( * , 1)). Let h = f g ∈ F \ K be given by the minimal admissible linear system A = (u, A, v) = (1, A, λ) of dimension n ≥ 2 and fix 1 ≤ k < n. Assume that A has a lower left block of zeros of size (n− k)× k) and an upper right block of zeros of size (k − 1) × (n − k). For a transformation (P, Q) let a ′ ij denote the entries of P AQ. If for each transformation (P, Q) of the form (4.1) respecting these zero blocks there exists an j ∈ {k + 1, k + 2, . . . , n} such that a ′ k,j is non-scalar then f is a left factor of type ( * , 1) of h with rank(f ) = k and rank(g) = n − k + 1.
Proof. By assumption, A is of the (block) form with square diagonal blocks A 1,1 , A 2,2 and A 3,3 of size k − 1, 1 and n − k respectively. We duplicate the entry t k in the right family by inserting a "dummy" column (and row) to get the following ALS of size n + 1: that is, "reversing" the construction from Proposition 2.11. The subsystems of dimension k and n−k +1 are minimal for f = µt k and g (due to the coupling condition) respectively, otherwise we could construct an ALS for h of dimension n ′ < n, contradicting minimality of A. Clearly, 1 ∈ R(g). Showing that f is a left factor of h = f g is like in Lemma 4.5.
Lemma 4.7 (Factorization Type (0, 0)). Let h = f g ∈ F \ K be given by the minimal admissible linear system A = (u, A, v) = (1, A, λ) of dimension n ≥ 2 and fix 1 ≤ k < n. If A = (a ij ) has a lower left block of zeros of size (n − k) × k, an upper right block of zeros of size k × (n − k − 1) and a i,k+1 ∈ K for i ∈ {1, 2, . . . , k} then f is a left factor of type ( * , 0) of h with rank(f ) = k and g is of type (0, * ) with rank(g) = n − k.
Proof. We get the subsystems A f (for f ) and A g (for g) directly from the construction of the multiplication in Proposition 2.1. Non-minimality of one of them would contradict minimality of A. As would 1 ∈ L(f ) or 1 ∈ R(g) using multiplication type ( * , 1) and (1, * ) respectively. The arguments for showing that f is left factor of h = f g are similar to that in (the proof of) Lemma 4.5.
Theorem 4.8 (Free Factorization). Let h ∈ F with n = rank(h) ≥ 2 be given by the minimal admissible linear system A = (u, A, v). Then h has a proper left factor f with rank(f ) = k if and only if there exists an admissible transformation (P, Q) of the form (4.1) such that P AQ is of "type" (1, * ), ( * , 1) or (0, 0) as in Figure 1.
Proof. Assuming a proper left factor of rank k, Lemma 4.3 applies. Conversely, assuming such a transformation, we get a proper left factor of rank k by Lemma 4.5 for type (1, * ), by Lemma 4.6 for type ( * , 1) and by Lemma 4.7 for type (0, 0).
The coupling conditions for type (0, 0) have to be implemented directly by adding the coefficients corresponding to x ∈ X for the "coupling vector". For type (1, * ) and ( * , 1) one can test for a "scalar" coupling first. If there is no solution one can try to find an appropriate transformation for the zero blocks only.
The first factor f 1 is an atom. Irreducibility of the second factor f 2 depends on the ground field K. Over K( X ) we have f 2 = (x − √ 2) −1 (x + √ 2) −1 .
Remark. To find a solution in general (more systematically), the primary decomposition of ideals can be used, see for example [CLO15,Section 4.8] and [Coh03a, Section 10.8].

Epilogue
The presented "free factorization theory" is concrete enough to be implemented in computer algebra software to be able to apply it. But beside a general concept for the construction of minimal linear representations (which is considered in future work), some more theoretical questions remain open: Is the extension of the "classical" factorization theory (in free associative algebras) to the free field -assuming that polynomial atoms (and their inverse) remain irreducibel-unique? Is the free field (in this setting) a "similarity UFD"? If so, given an element, is the sequence of the ranks of the atoms of a factorization an invariant (modulo permutations)?