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In this article, we prove that the product of two algebraic numbers of degrees 4 and 6 over Q{\mathbb{Q}} cannot be of degree 8. This completes the classification of so-called product-feasible triplets (a,b,c)∈N3\left(a,b,c)\in {{\mathbb{N}}}^{3} with a≤b≤ca\le b\le c and b≤7b\le 7. The triplet (a,b,c)\left(a,b,c) is called product-feasible if there are algebraic numbers α,β\alpha ,\beta , and γ\gamma of degrees a,ba,b, and cc over Q{\mathbb{Q}}, respectively, such that αβ=γ\alpha \beta =\gamma . In the proof, we use a proposition that describes all monic quartic irreducible polynomials in Q[x]{\mathbb{Q}}\left[x] with four roots of equal moduli and is of independent interest. We also prove a more general statement, which asserts that for any integers n≥2n\ge 2 and k≥1k\ge 1, the triplet (a,b,c)=(n,(n−1)k,nk)\left(a,b,c)=\left(n,\left(n-1)k,nk) is product-feasible if and only if nn is a prime number. The choice (n,k)=(4,2)\left(n,k)=\left(4,2) recovers the case (a,b,c)=(4,6,8)\left(a,b,c)=\left(4,6,8) as well.

For various types of field extensions E / F and values of n ∈ ℕ we consider quadratic forms φ that lie in the n th power I n E of the fundamental ideal of E but are already defined over F. We then search for some ψ ∈ I n F of minimal dimension that maps to φ when extending the scalars to E. This problem can be easily solved completely for finite extensions of odd degree. For quadratic extensions E / F , the situation is more involved, but solved for n ≤ 3. For n > 3 , we construct quadratic field extensions E / F and a form φ ∈ I n E such that any form ψ ∈ I n F that maps to φ when extending the scalars to E has dimension at least dim φ + 2 n − 2 . [ABSTRACT FROM AUTHOR]

Let Φ be a field of prime characteristic p and let G be a finite group. We develop an equivalence relation between the set of isomorphism types of indecomposable (simple) KG-modules, where K is any finite subfield of Φ , and relate the equivalence classes to the set of isomorphism types of indecomposable (resp. simple) Φ G -modules. When Φ is the algebraic closure of a field F of order p, we study indecomposable (resp. simple) Φ G - modules and obtain a classification of the isomorphism types of simple Φ G -modules and a new formula for the number of such types in each equivalence class. [ABSTRACT FROM AUTHOR]

We look at Hopf Galois structures with square free pqw degree on separable field extensions (nonnormal) L/K. Where E/K is the normal closure of L/K, the group permutation of degree pqw is G = Gal(E/K). We study details of the nonabelian case, where Jl = ⟨σ, [τ, αl]⟩ is a nonabelian regular subgroup of Hol(N) for 1 ≤ l ≤ w - 1. We first find the group permutation G, and then the Hopf Galois structures for each G. In this case, there exists four G such that the Hopf Galois structures are admissible within the field extensions L/K. [ABSTRACT FROM AUTHOR]

In 2020, Alabdali and Byott described the Hopf-Galois structures arising on Galois field extensions of squarefree degree. Extending to squarefree separable, but not necessarily normal, extensions L / K is a natural next step. One must consider now the interplay between two Galois groups G = Gal (E / K) and G ′ = Gal (E / L) , where E is the Galois closure of L / K. In this paper, we give a characterisation and enumeration of the Hopf-Galois structures arising on separable extensions of degree pq where p and q are distinct odd primes. This work includes the results of Byott and Martin-Lyons who do likewise for the special case that p = 2 q + 1. [ABSTRACT FROM AUTHOR]

Let K / F be a finite Galois extension of fields with Gal (K / F) = Γ. In an earlier work of Timothy Kohl, the author enumerated dihedral Hopf-Galois structures acting on dihedral extensions. The dihedral group is one particular example of a semidirect product of Z n and Z 2. In this article we count the number of Hopf-Galois structures with Galois group Γ of type G , where Γ , G are groups of the form Z n ⋊ ϕ Z 2 when n is odd with radical of n being a Burnside number. As an application we also find the corresponding number of skew braces. [ABSTRACT FROM AUTHOR]

All gradings of quadratic Kummer extensions by a group are described. These extensions have a natural grading by Galois group and its subgroups. Sufficient and necessary conditions for such extensions to have nontrivial gradings by other groups are determined. [ABSTRACT FROM AUTHOR]

In this article we further develop field theory [6], [7], [12] in Mizar [1], [2], [3]: we deal with quadratic polynomials and quadratic extensions [5], [4]. First we introduce quadratic polynomials, their discriminants and prove the midnight formula. Then we show that - in case the discriminant of p being non square - adjoining a root of p's discriminant results in a splitting field of p. Finally we prove that these are the only field extensions of degree 2, e.g. that an extension E of F is quadratic if and only if there is a non square Element a ∈ F such that E and ( F a F\sqrt a) are isomorphic over F. [ABSTRACT FROM AUTHOR]

Summary. In this article we further develop field theory in Mizar [1], [2]: we prove existence and uniqueness of splitting fields. We define the splitting field of a polynomial p ∈ F [X] as the smallest field extension of F, in which p splits into linear factors. From this follows, that for a splitting field E of p we have E = F (A) where A is the set of p's roots. Splitting fields are unique, however, only up to isomorphisms; to be more precise up to F -isomorphims i.e. isomorphisms i with i|F = IdF. We prove that two splitting fields of p ∈ F [X] are F -isomorphic using the well-known technique [4], [3] of extending isomorphisms from F1 → F2 to F1(a) → F2(b) for a and b being algebraic over F1 and F2, respectively. [ABSTRACT FROM AUTHOR]

In previous work (Dawsey et al. in J Algebra 533:339–343, 2019), the authors confirmed the speculation of J.G. Thompson that certain multiquadratic fields are generated by specified character values of sufficiently large alternating groups A n . Here we address the natural generalization of this speculation to the finite general linear groups GL m F q and SL 2 F q . [ABSTRACT FROM AUTHOR]

Given a collection A = { L 1 , ... , L n } of intermediate fields in a field extension L/K of finite degree and Λ L , A = L 1 × ⋯ × L n , there is a natural map Ψ L , A : L → Λ L , A given by y ↦ ( Tr L / L 1 (y) , ... , Tr L / L n (y)) , where Tr L / L i : L → L i denotes the trace. The image set Ψ L , A (L) turns out to be contained in a certain subset Λ L , A * of Λ L , A (in fact, a K-subspace). In this paper, we exhibit classes of extensions L/K where the equality Ψ L , A (L) = Λ L , A * holds for every collection A of intermediate fields in L/K, and we also give examples to the contrary. Finally we provide combinatorial formulas for the dimension of Λ L , A *. [ABSTRACT FROM AUTHOR]

Let n be a squarefree natural number, and let G , Γ be two groups of order n. We determine the number of Hopf-Galois structures of type G admitted by a Galois extension of fields with Galois group isomorphic to Γ. We give some examples, including a full treatment of the case where n is the product of three primes. [ABSTRACT FROM AUTHOR]

In [7], [9], [10] we presented a formalization of Kronecker's construction of a field extension E for a field F in which a given polynomial p ∈ F [X]\F has a root [5], [6], [3]. A drawback of our formalization was that it works only for polynomial-disjoint fields, that is for fields F with F ∩ F [X] = ∅. The main purpose of Kronecker's construction is that by induction one gets a field extension of F in which p splits into linear factors. For our formalization this means that the constructed field extension E again has to be polynomial-disjoint. In this article, by means of Mizar system [2], [1], we first analyze whether our formalization can be extended that way. Using the field of polynomials over F with degree smaller than the degree of p to construct the field extension E does not work: In this case E is polynomial-disjoint if and only if p is linear. Using F [X]/ one can show that for F = ℚ and F = ℤn the constructed field extension E is again polynomial-disjoint, so that in particular algebraic number fields can be handled. For the general case we then introduce renamings of sets X as injective functions f with dom(f) = X and rng(f) ∩ (X ∪ Z) = ∅ for an arbitrary set Z. This, finally, allows to construct a field extension E of an arbitrary field F in which a given polynomial p ∈ F [X]\F splits into linear factors. Note, however, that to prove the existence of renamings we had to rely on the axiom of choice. [ABSTRACT FROM AUTHOR]

This is the fourth part of a four-article series containing a Mizar [3], [2], [1] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/ as the desired field extension E [6], [4], [5]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ F [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in the second part the field (E \ ϕF)∪F for a given monomorphism ϕ: F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that "one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/". Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker's construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitrary fields F : With the exception of ℤ2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar's representations of ℤn, ℚ and ℝ we have ℤn ∩ ℤn[X] = ∅, ℚ ∩ ℚ[X] = ∅ and ℝ ∩ ℝ[X] = ∅, respectively. In this fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ: F → F [X]/. Together with the first part this gives – for fields F with F ∩ F [X] = ∅ – a field extension E of F in which p ∈ F [X]\F has a root. [ABSTRACT FROM AUTHOR]

This is the third part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/ as the desired field extension E [6], [4], [5]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ F [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in the second part the field (E \ ϕF)∪F for a given monomorphism ϕ: F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that "one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/". Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker's construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in this third part, this condition is not automatically true for arbitrary fields F : With the exception of ℤ2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar's representations of ℤn, ℚ and ℝ we have ℤn ∩ ℤn[X] = ∅, ℚ ∩ ℚ[X] = ∅ and ℝ ∩ ℝ[X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ: F → F [X]/. Together with the first part this gives – for fields F with F ∩ F [X] = ∅ – a field extension E of F in which p ∈ F [X]\F has a root. [ABSTRACT FROM AUTHOR]

In this short note, we answer a question raised by M. Papikian on a universal upper bound for the degree of the extension of K ∞ given by adjoining the periods of a Drinfeld module of rank 2. We show that contrary to the rank 1 case such a universal upper bound does not exist, and the proof generalies to higher rank. Moreover, we give an upper and lower bound for the extension degree depending on the valuations of the defining coefficients of the Drinfeld module. In particular, the lower bound shows the non-existence of a universal upper bound. [ABSTRACT FROM AUTHOR]

This is the second part of a four-article series containing a Mizar [2], [1] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/ as the desired field extension E [5], [3], [4]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ : F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in this second part the field (E \ ϕF)∪F for a given monomorphism ϕ : F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that "one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/". Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker's construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields F : With the exception of 𝕑2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar's representations of 𝕑n, 𝕈 and 𝕉 we have 𝕑n ∩ 𝕑n[X] = ∅, 𝕈 ∩ 𝕈 [X] = ∅ and 𝕉 ∩ 𝕉 [X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ : F → F [X]/. Together with the first part this gives - for fields F with F ∩ F [X] = ∅ - a field extension E of F in which p ∈ F [X]\F has a root. [ABSTRACT FROM AUTHOR]

This is the first part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/ as the desired field extension E [9], [4], [6]. In this first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/ and show that the translated polynomial ϕ(p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in the second part the field (E \ ϕF)∪F for a given monomorphism ϕ : F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that "one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/". Interestingly, to do so we need to assume that F ∩ E =∅, in particular Kronecker's construction can be formalized for fields F with F \ F [X] =∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields F : With the exception of 𝕑2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠∅. We also prove that for Mizar's representations of 𝕑n, 𝕈 and 𝕉 we have 𝕑n ∩ 𝕑n[X] = ∅, 𝕈 ∩ 𝕈[X] = ∅and 𝕉 ∩ 𝕉[X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F i F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ : F → F [X]/. Together with the first part this gives - for fields F with F ∩ F [X] = ∅ - a field extension E of F in which p ∈ F [X]\F has a root. [ABSTRACT FROM AUTHOR]

Abstract We classify all skew braces of Heisenberg type for a prime number p > 3. Furthermore, we determine the automorphism group of each one of these skew braces (as well as their socle and annihilator). Hence, by utilising a link between skew braces and Hopf–Galois theory, we can determine all Hopf–Galois structures of Heisenberg type on Galois field extensions of fields of degree p 3. [ABSTRACT FROM AUTHOR]

Recently, Bollen, Draisma, and Pendavingh have introduced the Lindström valuation on the algebraic matroid of a field extension of characteristic p . Their construction passes through what they call a matroid flock and builds on some of the associated theory of matroid flocks which they develop. In this paper, we give a direct construction of the Lindström valuated matroid using the theory of inseparable field extensions. In particular, we give a description of the valuation, the valuated circuits, and the valuated cocircuits. [ABSTRACT FROM AUTHOR]

We investigate Hopf–Galois structures on a cyclic field extension L / K of squarefree degree n . By a result of Greither and Pareigis, each such Hopf–Galois structure corresponds to a group of order n , whose isomorphism class we call the type of the Hopf–Galois structure. We show that every group of order n can occur, and we determine the number of Hopf–Galois structures of each type. We then express the total number of Hopf–Galois structures on L / K as a sum over factorisations of n into three parts. As examples, we give closed expressions for the number of Hopf–Galois structures on a cyclic extension whose degree is a product of three distinct primes. (There are several cases, depending on congruence conditions between the primes.) We also consider one case where the degree is a product of four primes. [ABSTRACT FROM AUTHOR]

An n × n matrix pattern is said to be spectrally arbitrary over a field F provided for every monic polynomial p ( t ) of degree n , with coefficients from F , there exists a matrix with entries from F , in the given pattern, that has characteristic polynomial p ( t ) . Let E ⊆ F ⊆ K be an extension of fields. It is natural to ask whether a pattern that is spectrally arbitrary over F must also be spectrally arbitrary over E or K . In this article it is shown that if F is dense in K and K is a complete metric space, then any spectrally arbitrary or relaxed spectrally arbitrary pattern over F is relaxed spectrally arbitrary over K . It is also established that if E is an algebraically closed subfield of a field F , then any spectrally arbitrary pattern over F is spectrally arbitrary over E . The 2 n Conjecture and the Superpattern Conjecture are explored over fields other than the real numbers. In particular, examples are provided to show that the Superpattern Conjecture is false over the field with 3 elements. [ABSTRACT FROM AUTHOR]

In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of ‘extra functional equations’ involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper, we generalize their result by proving a similar identity relating certain degree-$\ell$ fields to Galois groups $D_{\ell }$ and $F_{\ell }$, respectively, for any odd prime $\ell$; in particular, we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms. [ABSTRACT FROM AUTHOR]

Generalizing John Conway's construction of the Field On 2 , we give the “minimal” definitions of addition and multiplication that turn the ordinals into a Field of characteristic p , for any prime p . We then analyze the structure of the resulting Field, which we will call On p . [ABSTRACT FROM AUTHOR]

Let A be a finite-dimensional algebra over a finite field F q and let G = A × be the multiplicative group of A . In this paper, we construct explicitly a generic Galois G -extension S / R , where R is a localized polynomial ring over F q , and an explicit generic polynomial for G in dim F q ⁡ ( A ) parameters. [ABSTRACT FROM AUTHOR]

In 2021, Marco Besier and the first author introduced the concept of rationalizability of square roots to simplify arguments of Feynman integrals. In this work, we generalize the definition of rationalizability to field extensions. We then show that the rationalizability of a set of quadratic field extensions is equivalent to the rationalizability of the compositum of the field extensions, providing a new strategy to prove rationalizability of sets of square roots of polynomials. [ABSTRACT FROM AUTHOR]

We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions L / K of squarefree degree n. If E / K is the normal closure of L / K then G = Gal (E / K) can be viewed as a permutation group of degree n. We show that G has derived length at most 4, but that many permutation groups of squarefree degree and of derived length 2 cannot occur. We then investigate in detail the case where n = p q where q ≥ 3 and p = 2 q + 1 are both prime. (Thus q is a Sophie Germain prime and p is a safeprime). We list the permutation groups G which can arise, and we enumerate the Hopf-Galois structures for each G. There are six such G for which the corresponding field extensions L / K admit Hopf-Galois structures of both possible types. [ABSTRACT FROM AUTHOR]

Abstract: In a previous piece of work it has been shown that the exponential stability of a linear time-varying (LTV) system can be evaluated using new definitions of the poles of such a system. The latter are given by a fundamental set of roots of the skew polynomial which defines the autonomous part of the system. Such a set may not exist over the initial field of definition of the coefficients of the system, but can exist over a suitable field extension . It is shown here that conditions for stability can also be obtained using linear factors of the polynomial over another field extension which may be smaller: . The roots of these factors are called the quasi-poles of the system. The necessary condition for system stability, expressed in function of these quasi-poles, is more restrictive than the one involving a fundamental set of roots. [Copyright &y& Elsevier]

Let F be a non-Archimedean local field and let E be a finite extension of F. Let G be an F-split semisimple F-group. We discuss how to compare volumes on the Bruhat-Tits buildings BE and BF of G(E) and G(F) respectively. [ABSTRACT FROM AUTHOR]

Let B and B be disjoint separable algebraic extensions of a field F, and let B = B B be their composite. Let α be an element of B and α be an element of B. Suppose α and α are primitive (resp. normal, resp. subnormal). We investigate the question of when α + α and α α are necessarily primitive (resp. normal, resp. subnormal) elements of B. (A normal element of a Galois extension is defined to be one that is part of a normal basis, and a subnormal element is defined analogously for a non-Galois extension). [ABSTRACT FROM AUTHOR]

Nonvanishing determinants have emerged as an attractive criterion enabling a space-time code achieve the optimal diversity-multiplexing gains tradeoff. It seems that cyclic division algebras play the most crucial role in designing a space-time code with nonvanishing determinants. In this paper, we explicitly construct space-time codes for arbitrary numbers of transmit antennas that achieve nonvanishing determinants and the optimal diversity-multiplexing gains tradeoff over Z [i]. Unlike previous methods usually arising a field compositum for two or more fields, our scheme, which only requires one simple extension, constitutes a much more efficient and feasible advancement whether in theory or practice. [ABSTRACT FROM AUTHOR]

Let K ⊂ R ⊂ P be a tower of fields, N be a P-module, and Δ: R → N be a K-linear differential operator. The aim of this paper is to investigate whether the operator Δ has an extension to P, i.e. if these exists a differential operator Δ′: P → N such that Δ′|R = Δ. The results of this paper were published in Russian in Mat. Zametki 30(2) (1981), 237–248. [ABSTRACT FROM AUTHOR]