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Let A = F q [ T ] be the polynomial ring over F q , and F be the field of fractions of A. Let ϕ be a Drinfeld A-module of rank r ≥ 2 over F. For all but finitely many primes p ◁ A , one can reduce ϕ modulo p to obtain a Drinfeld A-module ϕ ⊗ F p of rank r over F p = A / p . The endomorphism ring E p = End F p ( ϕ ⊗ F p ) is an order in an imaginary field extension K of F of degree r. Let O p be the integral closure of A in K, and let π p ∈ E p be the Frobenius endomorphism of ϕ ⊗ F p . Then we have the inclusion of orders A [ π p ] ⊂ E p ⊂ O p in K. We prove that if End F alg ( ϕ ) = A , then for arbitrary non-zero ideals n , m of A there are infinitely many p such that n divides the index χ ( E p / A [ π p ] ) and m divides the index χ ( O p / E p ) . We show that the index χ ( E p / A [ π p ] ) is related to a reciprocity law for the extensions of F arising from the division points of ϕ. In the rank r = 2 case we describe an algorithm for computing the orders A [ π p ] ⊂ E p ⊂ O p , and give some computational data.

Let G be a connected simple graph with characteristic polynomial P G ( x ) . The irreducibility of P G ( x ) over rational numbers Q has a close relationship with the automorphism group, reconstruction and controllability of a graph. In this paper we derive three methods to construct graphs with irreducible characteristic polynomials by appending paths P 2 n + 1 − 2 ( n ≥ 1 ) to certain vertices; union and join K 1 alternately and corona. These methods are based on Eisenstein's criterion and field extensions. Concrete examples are also supplied to illustrate our results.

For a given field F of characteristic different from 2 and $$a,b,d\in F^*$$ we construct an invariant $$\mathrm{inv}$$ for an element $$D\in \,_2\mathrm{Br}(F(\sqrt{a},\sqrt{b},\sqrt{d})/F)$$ . This invariant takes value in the quotient group $$\begin{aligned} H^3(F,\mu _2)/D\cup {\mathrm{N}_{\mathrm{F}\left( \sqrt{\mathrm{d}}, \sqrt{\mathrm{ab}}\right) /\mathrm{F}}}F\left( \sqrt{d},\sqrt{ab}\right) ^*. \end{aligned}$$ Let k be a field, let $$k(\sqrt{a},\sqrt{b},\sqrt{d})/k$$ be a triquadratic field extension. We apply the invariant $$\mathrm{inv}$$ and a few deep results from algebraic geometry and K-theory to construct a field extension K/k with $$\mathrm{cd}_2 K=3$$ , and an indecomposable cross product algebra of exponent 2 with respect to the extension $$K(\sqrt{a},\sqrt{b},\sqrt{d})/K$$ . Using the invariant $$\mathrm{inv}$$ , we also prove the following odd degree descent statement: Assume $$D\in \,_2\mathrm{Br}(F)$$ , $$b,d\in F^*$$ , L/F is an odd degree extension. Assume also that $$D_{L(\sqrt{b},\sqrt{d})}=Q_{L(\sqrt{b},\sqrt{d})}$$ , where Q is a quaternion algebra defined over L. Then there exists a quaternion algebra $$\widetilde{Q}$$ defined over F such that $$D_{F(\sqrt{b},\sqrt{d})}=\widetilde{Q}_{F(\sqrt{b},\sqrt{d})}$$ . As a consequence we get that if $$\phi \in I^2(F)$$ is a form such that $${(\phi _{L(\sqrt{b},\sqrt{d})})}_{an}$$ is defined over L, and $$\dim {(\phi _{L(\sqrt{b},\sqrt{d})})}_{an} =4$$ , then $${(\phi _{F(\sqrt{b},\sqrt{d})})}_{an}$$ is defined over F.

Let $k$ be a field with char $k\neq 2$ and $k$ be not algebraically closed. Let $a\in k\setminus k^2$ and $L=k(\sqrt{a})(x,y)$ be a field extension of $k$ where $x,y$ are algebraically independent over $k$. Assume that $\sigma$ is a $k$-automorphism on $L$ defined by \[ \sigma: \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c(x+\frac{b}{x})+d}{y} \] where $b,c,d \in k$, $b\neq 0$ and at least one of $c,d$ is non-zero. Let $L^{\langle\sigma\rangle}=\{u\in L:\sigma(u)=u\}$ be the fixed subfield of $L$. We show that $L^{\langle\sigma\rangle}$ is isomorphic to the function field of a certain surface in $P^4_k$ which is given as the intersection of two quadrics. We give criteria for the $k$-rationality of $L^{\langle\sigma\rangle}$ by using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Th\'el\`ene., Comment: To appear in Manuscripta Math. The main theorems (old Theorem 1.7 and Theorem 1.8) incorporated into (new) Theorem 1.8. Section 3 and Section 4 interchanged

Let K be a finitely generated extension of a field k of characteristic $$p\not =0$$ . In 1947, Dieudonne initiated the study of maximal separable intermediate fields. He gave in particular the form of an important subclass of maximal separable intermediate fields D characterized by the property $$K\subseteq k({D}^{p^{-\infty }})$$ , and which are called the distinguished subfields of K/k. In 1970, Kraft showed that the distinguished maximal separable subfields are precisely those over which K is of minimal degree. This paper grew out of an attempt to find a new characterization of distinguished subfields of K/k by means of new inseparability invariants.

We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of the field with q elements. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code C, 14 pages

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K × . We consider Kummer extensions of G of the form K ( ζ 2 m , G 2 n ) / K ( ζ 2 m ) , where n ⩽ m . In the paper by Debry and Perucca (2016) [1] , the degrees of those extensions have been evaluated in terms of divisibility parameters over K ( ζ 4 ) . We prove how properties of G over K explicitly determine the divisibility parameters over K ( ζ 4 ) . This result yields a clear computational advantage, since no field extension is required.

Consider the abelian category ${\mathcal C}$ of commutative group schemes of finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp. ${\rm Pro}({\mathcal F})$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group $\varpi_1 : {\rm Pro}({\mathcal C}) \to {\rm Pro}({\mathcal F})$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors $\varpi_i$ vanish for $i \geq 2$. Along the way, we describe the indecomposable projective objects of ${\rm Pro}({\mathcal C})$ over an arbitrary field $k$., Final version, accepted for publication at Annales Henri Lebesgue

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in B α ∈ L [ y ] , where L = Q ( α ) is a simple algebraic extension of the rational numbers. We revisit two approaches for the problem. In the first approach, using resultant computations, we perform a reduction to a polynomial with integer coefficients and we deduce a bound of O ˜ B ( N 8 ) for isolating the real roots of B α , where N is an upper bound on all the quantities (degree and bitsize) of the input polynomials. The bound becomes O ˜ B ( N 7 ) if we use Pan's algorithm for isolating the real roots. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for the roots and we prove that they are optimal, under mild assumptions. For isolating the real roots we consider a modified Sturm algorithm, and a modified version of descartes ' algorithm. For the former we prove a Boolean complexity bound of O ˜ B ( N 12 ) and for the latter a bound of O ˜ B ( N 5 ) . We present aggregate separation bounds and complexity results for isolating the real roots of all polynomials B α k , when α k runs over all the real conjugates of α. We show that we can isolate the real roots of all polynomials in O ˜ B ( N 5 ) . Finally, we implemented the algorithms in C as part of the core library of MATHEMATICA and we illustrate their efficiency over various data sets.

Let A be a biquaternion division algebra over a field F ( char F ≠ 2 ). We prove that an element e ∈ F ⁎ such that A F ( e 4 ) = 0 exists if and only if there is a decomposition A ≃ ( a , b ) ⊗ ( c , d ) with 《 − a , b , c , d 》 = 0 . This statement strengthens the result of Rost, Serre and Tignol ( [7] ), where the additional condition − 1 ∈ F ⁎ holds. Another result is a construction of a field F with − 1 ∈ F ⁎ , a quartic cyclic field extension L / F , and quaternion algebras ( a , b ) and ( c , d ) over F such that ( a , b ) L = ( c , d ) L = 0 , but the quadratic form 〈 a , b , a b 〉 ⊥ 《 c , d 》 is anisotropic. As a consequence, for any algebraically closed field k 0 , we give an example of an anisotropic 7-dimensional quadratic form Ψ over k 0 ( z , x , y ) with ind ( C 0 ( Ψ ) ) = 4 such that for any discrete k 0 ( z ) -valuation v on k 0 ( z , x , y ) the form Ψ becomes isotropic over the completion k 0 ( z , x , y ) v . In other words, the form Ψ provides a counterexample to the strong Hasse principle for 7-dimensional quadratic forms with respect to the rational extension K ( x , y ) / K , where K = k 0 ( z ) .

We study orbit codes in the field extension \begin{document}$ \mathbb{F}_{q^n} $\end{document} . First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of \begin{document}$ \mathbb{F}_{q^n} $\end{document} . We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.

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 $G=\mathrm{Gal}(E/K)$ 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 $n=pq$ where $q \geq 3$ and $p=2q+1$ 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., Comment: 26 pages. Proposition 4.12 and Table 5 corrected. Theorem 3.5 removed. Typos fixed and some improvements made to wording. Accepted for Journal of Pure and Applied Algebra

We give a geometric interpretation of Broussous-Grabitz embedding types. We fix a central division algebra $D$ of finite index over a non-Archimedean local field $F$ and a positive integer $m$. Further we fix a hereditary order $\mathfrak{a}$ of $\operatorname{M}_m(D)$ and an unramified field extension $E|F$ in $\operatorname{M}_m(D)$ which is embeddable in $D$ and which normalizes $\mathfrak{a}$. Such a pair $(E,\mathfrak{a})$ is called an embedding. The embedding types classify the $\operatorname{GL}_m(D)$-conjugation classes of these embeddings. Such a type is a class of matrices with non-negative integer entries. We give a formula which allows us to recover the embedding type of $(E,\mathfrak{a})$ from the simplicial type of the image of the barycenter of $\mathfrak{a}$ under the canonical isomorphism, from the set of $E^\times$-fixed points of the reduced building of $\operatorname{GL}_m(D)$ to the reduced building of the centralizer of $E^\times$ in $\operatorname{GL}_m(D)$. Conversely the formula allows to calculate the simplicial type up to cyclic permutation of the Coxeter diagram.

Let k be a non-archimedean local field with residual characteristic p. Let G be a connected reductive group over k that splits over a tamely ramified field extension of k. Suppose p does not divide the order of the Weyl group of G. Then we show that every smooth irreducible complex representation of G(k) contains an $\mathfrak{s}$-type of the form constructed by Kim and Yu and that every irreducible supercuspidal representation arises from Yu's construction. This improves an earlier result of Kim, which held only in characteristic zero and with a very large and ineffective bound on p. By contrast, our bound on p is explicit and tight, and our result holds in positive characteristic as well. Moreover, our approach is more explicit in extracting an input for Yu's construction from a given representation., Comment: 39 pages, accepted for publication in Annals of Mathematics

In this essay, we study various notions of projective space (and other schemes) over F(l)e, with F-1 denoting the field with one element. Our leading motivation is the "Hidden Points Principle," which shows a huge deviation between the set of rational points as closed points defined over F-l(e) and the set of rational points defined as morphisms Spec(F-1(l)) (sic) X. We also introduce, in the same vein as Kurokawa [Proc. Jpn. Acad. Ser. A Math. Sci. 81 (2005), pp. 180-184], schemes of IF -type and consider their zeta functions.

Let K be the field of fractions of a complete discrete valuation ring $$ \mathcal{A} $$ with residue field k, and let G be a connected reductive algebraic group over K. Suppose $$ \mathcal{P} $$ is a parahoric group scheme attached to G. In particular, $$ \mathcal{P} $$ is a smooth affine $$ \mathcal{A} $$-group scheme having generic fiber $$ \mathcal{P} $$K = G; the group scheme $$ \mathcal{P} $$ is in general not reductive over $$ \mathcal{A} $$. If G splits over an unramified extension of K, we find in this paper a closed and reductive $$ \mathcal{A} $$-subgroup scheme $$ \mathcal{M}\subset \mathcal{P} $$ for which the special fiber $$ \mathcal{M} $$k is a Levi factor of $$ \mathcal{P} $$k. Moreover, we show that the generic fiber $$ M={\mathcal{M}}_{\mathrm{K}} $$ is a subgroup of G which is geometrically of type C(μ) – i.e., after a separable field extension, M is the identity component $$ M={C}_G^o\left(\phi \right) $$ of the centralizer of the image of a homomorphism ϕ: μn → H, where μn is the group scheme of n-th roots of unity for some n ≥ 2. For a connected and split reductive group H over any field $$ \mathcal{F} $$, the paper describes those subgroups of H which are of type C(μ).

Let $G$ be a group. Two elements $x,y \in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. Consider $\mathbb F$ a perfect field of characteristic $\neq 2$, which has a non-trivial Galois automorphism of order $2$. Further, suppose that the fixed field $\mathbb F_0$ has the property that it has only finitely many field extensions of any finite degree. In this paper, we prove that the number of $z$-classes in the unitary group over such fields is finite. Further, we count the number of $z$-classes in the finite unitary group $U_n(q)$, and prove that this number is same as that of $GL_n(q)$ when $q>n$., In section 2 Propositions 2.1, 2.2 are added and section 5 is added, final version

Motivated by the analysis of neural net models between integer and rational weights, we introduce a so-called cut language over a real digit alphabet, which contains finite β-expansions (i.e. base-β representations) of the numbers less than a given threshold. We say that an infinite β-expansion is eventually quasi-periodic if its tail sequence formed by the numbers whose representations are obtained by removing leading digits, contains an infinite constant subsequence. We prove that a cut language is regular iff its threshold is a quasi-periodic number whose all β-expansions are eventually quasi-periodic, by showing that altogether they have a finite number of tail values. For algebraic bases β, we prove that there is an eventually quasi-periodic β-expansion with an infinite number of tail values iff there is a conjugate of β on the unit circle. For transcendental β combined with algebraic digits, a β-expansion is eventually quasi-periodic iff it has a finite number of tail values. For a Pisot base β and digits from the smallest field extension Q ( β ) over rational numbers including β, we show that any number from Q ( β ) is quasi-periodic. In addition, we achieve a dichotomy that a cut language is either regular or non-context-free and we show that any cut language with rational parameters is context-sensitive.

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.

We analyze the rel leaves of the Arnoux-Yoccoz translation surfaces. We show that for any genus $g \geq 3$, the leaf is dense in the connected component of the stratum $H(g -1 , g -1)$ to which it belongs, and the one-sided imaginary-rel trajectory of the surface is divergent. For one surface on this trajectory, namely the Arnoux-Yoccoz surface itself, the horizontal foliation is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but for all other surfaces, the horizontal foliation is completely periodic. The appendix proves a field theoretic result needed for denseness of the leaf: for any $n \geq 3$, the field extension of the rationals obtained by adjoining a root of $X^n-X^{n-1}-\ldots-X-1$ has no totally real subfields other than the rationals., Comment: Appendix by Lior Bary-Soroker, Mark Shusterman and Umberto Zannier. The prior version was published, but had errors in \S 6. Erroneous statements have been indicated and an erratum was included as Appendix B which corrects the errors. Main results are still correct. 70 pages, 9 figures. arXiv admin note: text overlap with arXiv:1506.06773

The homology groups h1(l/k), h2(l/k), and h3(l/k) of the Brauer complex for a triquadratic field extension l=k(a,b,c) are studied. In particular, given D∈APTARAMOPREFIX2 Br (k(a,b,c)/k), we find equivalent conditions for the image of D in h2(l/k) to be zero. We consider as well the second divided power operation γ2:APTARAMOPREFIX2 Br (l/k)→H4(k,Z/2Z), and show that there are nonstandard elements with respect to γ2. Further, a natural transformation h2⊗h1→H3, which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension F/k such that the cohomology group h1(F(a,b,c)/F) of the Brauer complex contains the images of prescribed elements of k∗, provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions L/F with nontrivial h3(L/F) are given. As a consequence we show that the homology group h3(L/F) can be arbitrarily big.

We show that Rost nilpotence holds for a geometrically integral threefold X over a field k of characteristic 0 if and only if αk(X)*∘N( CH 0(Xk(X)))=0 for some integer N>0 for all correspondences α of degree 0 which vanish over some field extension of k. As a corollary we get the Rost nilpotence property for three-dimensional smooth projective geometrically integral schemes over a field of characteristic zero, which are birationally isomorphic to a toric model.

Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form $$\mathcal {E}_{A,B}:\, A(x)-B(y)=0$$ , where $$A, B\in \mathbb {C}(z)$$ . We also investigate “series” of curves $$\mathcal {E}_{A,B}$$ of genus zero, where by a series we mean a family with the “same” A. We show that for a given rational function A a sequence of rational functions $$B_i$$ , such that $$\deg B_i\rightarrow \infty $$ and all the curves $$A(x)-B_i(y)=0$$ are irreducible and have genus zero, exists if and only if the Galois closure of the field extension $$\mathbb {C}(z)/\mathbb {C}(A)$$ has genus zero or one.

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 2n 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.

In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) $A = (A_{\alpha \beta} x_{\alpha \beta})$, where $A_{\alpha \beta}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$ and $x_{\alpha \beta}$ is an indeterminate for $\alpha = 1,2,\dots, \mu$ and $\beta = 1,2, \dots, \nu$. This problem can be viewed as an algebraic generalization of the bipartite matching problem and was considered by Iwata and Murota (1995). Recent interests in this problem lie in the connection with non-commutative Edmonds' problem by Ivanyos, Qiao, and Subrahamanyam (2018) and Garg, Gurvits, Oliveiva, and Wigderson (2019), where a result by Iwata and Murota implicitly states that the rank and non-commutative rank (nc-rank) are the same for this class of symbolic matrices. The main result of this paper is a simple and combinatorial $O((\mu \nu)^2 \min \{ \mu, \nu \})$-time algorithm for computing the symbolic rank of a $(2 \times 2)$-type generic partitioned matrix of size $2\mu \times 2\nu$. Our algorithm is inspired by the Wong sequence algorithm by Ivanyos, Qiao, and Subrahamanyam for the nc-rank of a general symbolic matrix, and requires no blow-up operation, no field extension, and no additional care for bounding the bit-size. Moreover it naturally provides a maximum rank completion of $A$ for an arbitrary field $\mathbf{F}$., Comment: This is a post-peer-review, pre-copyedit version of an article published in Mathematical Programming. The final authenticated version is available online at: https://doi.org/10.1007/s10107-021-01676-5

In connection with Galois Theory and Algebraic Curves, this paper investigates rational functions $h(x) = f(x)/g(x) \in \mathbb{F}_q(x)$ for which the value set $V_h = {\{h(α) | α \in \mathbb{F}_q \cup\{\infty\}}\}$ is relatively small. In particular, under certain circumstances, it proves that $h(x)$ having a small value set is equivalent to the field extension $\mathbb{F}_q(x)/\mathbb{F}_q(h(x))$ being Galois., Oberwolfach Preprints;2020,05

A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the q-polynomial over $${{\mathbb {F}}}_{q^6}$$ F q 6 , $$q \equiv 1\pmod 4$$ q ≡ 1 ( mod 4 ) described in Bartoli et al. (ARS Math Contemp 19:125–145, 2020) and Zanella and Zullo (Discrete Math 343:111800, 2020) is generalized for any even $$n\ge 6$$ n ≥ 6 to an $${{{\mathbb {F}}}_q}$$ F q -linear automorphism $$\psi (x)$$ ψ ( x ) of $${{\mathbb {F}}}_{q^n}$$ F q n of order n. Such $$\psi (x)$$ ψ ( x ) and some functional powers of it are proved to be scattered. In particular, this provides new maximum scattered linear sets of the projective line $${{\,\mathrm{{PG}}\,}}(1,q^n)$$ PG ( 1 , q n ) for $$n=8,10$$ n = 8 , 10 . The polynomials described in this paper lead to a new infinite family of MRD-codes in $${{\mathbb {F}}}_q^{n\times n}$$ F q n × n with minimum distance $$n-1$$ n - 1 for any odd q if $$n\equiv 0\pmod 4$$ n ≡ 0 ( mod 4 ) and any $$q\equiv 1\pmod 4$$ q ≡ 1 ( mod 4 ) if $$n\equiv 2\pmod 4$$ n ≡ 2 ( mod 4 ) .

Let $n$ be a squarefree natural number, and let $G$, $\Gamma$ 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 $\Gamma$. We give some examples, including a full treatment of the case where $n$ is the product of three primes., Comment: 29 pages; arXiv identifier of our preprint on skew braces now included

In this short note, we answer a question raised by M. Papikian on a universal upper bound for the degree of the extension of $K_\infty$ 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 generalises 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., 7 pages; v1->v2: corrected typos, added/changed references, now 8 pages

Let G be a finite group, N and H subgroups of G with N

Let G be a split simple affine algebraic group of type A or C over a field k, and let E be a standard generic G-torsor over a field extension of k. We compute the Chow ring of the variety of Borel subgroups of G (also called the variety of complete flags of G), twisted by E. In most cases, the answer contains a large finite torsion subgroup. The torsion-free cases have been treated in the predecessor Chow ring of some generically twisted flag varieties by the author.

Let \(k_0\) be a field of characteristic distinct from 2, \(l_0{/}k_0\) a finite field extension of degree \(2^m\), \(m\ge 2\). We prove that there exists a field extension \(K{/}k_0\) linearly disjoint with \(l_0/k_0\) such that the extension \(l_0K{/}K\) is nonexcellent. The crucial point is a construction of in some sense nonstandard elements in \(H^3(K,\mathbb {Z}{/}2\mathbb {Z})\). We apply this construction as well for investigation of the group \({F^*}^2N_{L{/}F}L^*\), where L / F is a \(\mathbb {Z}{/}4\mathbb {Z}\times \mathbb {Z}{/}2\mathbb {Z}\)-Galois extension. More precisely, let \(F\subset F_i\subset L\)\((1\le i\le 3)\) be the three intermediate quadratic extensions of F, and \(N_i(F)=N_{F_i{/}F}F_i^*\). We show that the quotient group \({N_1(F)\cap N_2(F)\cap N_3(F)\over {F^*}^2N_{L/F}L^*}\) can be arbitrarily large.

For $\Gamma$ a group of order $mp$ for $p$ prime where $gcd(p,m)=1$, we consider those regular subgroups $N\leq Perm(\Gamma)$ normalized by $\lambda(\Gamma)$, the left regular representation of $\Gamma$. These subgroups are in one-to-one correspondence with the Hopf-Galois structures on separable field extensions $L/K$ with $\Gamma=Gal(L/K)$. This is a follow up to the author's earlier work where, by assuming $p>m$, one has that all such $N$ lie within the normalizer of the $p$-Sylow subgroup of $\lambda(\Gamma)$. Here we show that one only need assume that all groups of a given order $mp$ have a unique $p$-Sylow subgroup, and that $p$ not be a divisor of the automorphism groups of any group of order $m$. As such, we extend the applicability of the program for computing these regular subgroups $N$ and concordantly the corresponding Hopf-Galois structures on separable extensions of degree $mp$.

The work of Greither and Pareigis details the enumeration of the Hopf-Galois structures (if any) on a given separable field extension. For an extension L / K which is classically Galois with G = G a l ( L / K ) the Hopf algebras in question are of the form ( L [ N ] ) G where N ≤ B = P e r m ( G ) is a regular subgroup that is normalized by the left regular representation λ ( G ) ≤ B . We consider the case where both G and N are isomorphic to a dihedral group D n for any n ≥ 3 . Using the normal block systems inherent to the left regular representation of each D n , (and every other regular permutation group isomorphic to D n ) we explicitly enumerate all possible such N which arise.

In this paper we develop a Mizar formalization of Kronecker's construction, which states that for every field $F$ and irreducible polynomial $p \in F[X]$ there exists a field extension $E$ of $F$ such that $p$ has a root over $E$ . It turns out that to prove the correctness of the construction the field $F$ needs to provide a disjointness condition, namely $F \cap F[X] = \emptyset$ . Surprisingly this property does not hold for arbitrary representations of a field $F{:}$ We construct for almost every field $F$ another representation $F^{\prime}$ , i.e. an isomorphic copy $F^{\prime}$ of $F$ , not satisfying this condition. As a consequence to $F^{\prime}$ our formalization of Kronecker's construction cannot be applied. All proofs have been carried out in the Mizar system. Based on Mizar's representation of the fields $\mathbb{Z}_{p}, \mathbb{Q}$ , and $\mathbb{R}$ we also have proven that $\mathbb{Z}_{p}\cap \mathbb{Z}_{p}[X]=\emptyset,\ \mathbb{Q}\cap\mathbb{Q}[X] = \emptyset$ , and $\mathbb{R}\cap \mathbb{R}[X]=\emptyset$ respectively.

A Theorem of Hou, Leung and Xiang generalised Kneser's addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give an alternative proof of the theorem that also holds in the non-separable case, thus solving Hou's conjecture. This result is a consequence of a strengthening of Hou et al.'s theorem that is a transposition to extension fields of an addition theorem of Balandraud., 17 pages

Let k be a base field, K be a field containing k and L/K be a field extension of degree n. The essential dimension ed(L/K) over k is a numerical invariant measuring "the complexity" of L/K. Of particular interest is $\tau$(n) = max { ed(L/K) | L/K is a separable extension of degree n}, also known as the essential dimension of the symmetric group $S_n$. The exact value of $\tau$(n) is known only for n $\leq$ 7. In this paper we assume that k is a field of characteristic p > 0 and study the essential dimension of inseparable extensions L/K. Here the degree n = [L:K] is replaced by a pair (n, e) which accounts for the size of the separable and the purely inseparable parts of L/K respectively, and \tau(n) is replaced by $\tau$(n, e) = max { ed(L/K) | L/K is a field extension of type (n, e)}. The symmetric group $S_n$ is replaced by a certain group scheme $G_{n,e}$ over k. This group is neither finite nor smooth; nevertheless, computing its essential dimension turns out to be easier than computing the essential dimension of $S_n$. Our main result is a simple formula for \tau(n, e)., Comment: 18 pages

Let $A=\mathbb{F}_q[T]$ be the polynomial ring over $\mathbb{F}_q$, and $F$ be the field of fractions of $A$. Let $\phi$ be a Drinfeld $A$-module of rank $r\geq 2$ over $F$. For all but finitely many primes $\mathfrak{p}\lhd A$, one can reduce $\phi$ modulo $\mathfrak{p}$ to obtain a Drinfeld $A$-module $\phi\otimes\mathbb{F}_\mathfrak{p}$ of rank $r$ over $\mathbb{F}_\mathfrak{p}=A/\mathfrak{p}$. The endomorphism ring $\mathcal{E}_\mathfrak{p}=\mathrm{End}_{\mathbb{F}_\mathfrak{p}}(\phi\otimes\mathbb{F}_\mathfrak{p})$ is an order in an imaginary field extension $K$ of $F$ of degree $r$. Let $\mathcal{O}_\mathfrak{p}$ be the integral closure of $A$ in $K$, and let $\pi_\mathfrak{p}\in \mathcal{E}_\mathfrak{p}$ be the Frobenius endomorphism of $\phi\otimes\mathbb{F}_\mathfrak{p}$. Then we have the inclusion of orders $A[\pi_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p}\subset \mathcal{O}_\mathfrak{p}$ in $K$. We prove that if $\mathrm{End}_{F^\mathrm{alg}}(\phi)=A$, then for arbitrary non-zero ideals $\mathfrak{n}, \mathfrak{m}$ of $A$ there are infinitely many $\mathfrak{p}$ such that $\mathfrak{n}$ divides the index $\chi(\mathcal{E}_\mathfrak{p}/A[\pi_\mathfrak{p}])$ and $\mathfrak{m}$ divides the index $\chi(\mathcal{O}_\mathfrak{p}/\mathcal{E}_\mathfrak{p})$. We show that the index $\chi(\mathcal{E}_\mathfrak{p}/A[\pi_\mathfrak{p}])$ is related to a reciprocity law for the extensions of $F$ arising from the division points of $\phi$. In the rank $r=2$ case we describe an algorithm for computing the orders $A[\pi_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p}\subset \mathcal{O}_\mathfrak{p}$, and give some computational data., Comment: Journal of Number Theory (David Goss Memorial Issue), to appear

By a result of Kantor, any subgroup of GL(n, q) containing a Singer cycle normalizes a field extension subgroup. This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PΓL(n, q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s, qs) under the canonical homomorphism GL(n, q) → PGL(n, q), for some divisor s of n, and so is contained in the image of ΓL(n/s, qs) under the canonical homomorphism ΓL(n, q) → PΓL(n, q). The actions of field extension subgroups on V (n, q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s, qs) of GL(n, q) has a unique orbit on s-dimensional subspaces of V (n, q) of length coprime to q. This orbit is a Desarguesian s-partition of V (n, q).

Let k be a field, K/k be a quadratic separable field extension, and 𝒜 a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically closed residue field of characteristic zero and the degree of 𝒜 is odd, we prove that all K/k-involutions on 𝒜 are cyclic.

Suppose E is an elliptic curve over $\Bbb Q$, and p>3 is a split multiplicative prime for E. Let q ≠ p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur–Tate–Teitelbaum conjecture for E at the prime p, over number fields $K\subset \Bbb Q\big(\mu_{{q^{\infty}}},\;\!^{q^{\infty}\!\!\!\!}\sqrt{m}\big)$ such that p remains inert in $K\cap\Bbb Q(\mu_{{q^{\infty}}})^+$. The proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.

Let k be field of characteristic ≠ 2, 3. Assume that k contains more than 11 elements and let K be an algebraic field extension of k. The present paper describes subgroups of the general linear group GL 7 ( K ) that contain the Chevalley group G 2 ( k ) .

A general theorem on the purity of invariant field extensions is proved. Using it, a criterion of rational triangulability of connected solvable affine algebraic subgroups of the Cremona groups is obtained. This criterion is applied for proving the existence of rationally nontriangulable subgroups of the above form and for proving their stable rational triangulability. The latter property answers in the affirmative Bass’ Triangulability Problem in the stable range. A general construction of all rationally triangulable connected solvable affine algebraic subgroups of the Cremona groups is obtained. As an application, a classification of all rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy is given.

Let $k$ be a field, $n \geqslant 5$ be an integer, $x_1, \dots, x_n$ be independent variables and $L_n = k(x_1, \dots, x_n)$. The symmetric group $S_n$ acts on $L_n$ by permuting the variables, and the projective linear group ${\rm PGL}_2$ acts by applying (the same) fractional linear transformation to each varaible. The fixed field $L_n^{{\rm PGL}_2}$ is called "the field of cross-ratios". Let $S \subset S_n$ be a subgroup. The Noether Problem asks whether the field extension $L_n^S/k$ is rational, and the Noether Problem for cross-ratios asks whether $K_n^S/k$ is rational. In an effort to relate these two problems, H. Tsunogai posed the following question: Is $L_n^S$ rational over $K_n^S$? He answered this question in several situations, in particular, in the case where $S = S_n$. In this paper we extend his results by recasting the problem in terms of Galois cohomology. Our main theorem asserts that the following conditions on a subgroup $S \subset S_n$ are equivalent: (a) $L_n^S$ is rational over $K_n^S$, (b) $L_n^S$ is unirational over $K_n^S$, (c) $S$ has an orbit of odd order in $\{1, \dots, n \}$., Comment: 7 pages

We show that each real-representable matroid is a minor of a complex-representable excluded minor for real-representability. More generally, for an infinite field $\mathbb{F}_1$ and a field extension $\mathbb{F}_2$, if $\mathbb{F}_1$-representability is not equivalent to $\mathbb{F}_2$-representability, then each $\mathbb{F}_1$-representable matroid is a minor of a $\mathbb{F}_2$-representable excluded minor for $\mathbb{F}_1$-representability., Comment: 14 pages, 2 figures. Submitted for publication in JCTb

Let $k$ be a field, with absolute Galois group $\Gamma$. Let $A/k$ be a finite \'etale group scheme of multiplicative type, i.e. a discrete $\Gamma$-module. Let $n \geq 2$ be an integer, and let $x \in H^n(k,A)$ be a cohomology class. We show that there exists a countable set $I$, and a familiy $(X_i)_{i \in I}$ of (smooth, geometrically integral) $k$-varieties, such that the following holds. For any field extension $l/k$, the restriction of $x$ vanishes in $H^n(l,A)$ if and only if (at least) one of the $X_i$'s has an $l$-point. We moreover show that the $X_i$'s can be made into an ind-variety. In the case $n=2$, we note that one variety is enough., Comment: 13 pages

We are interested in characterising pairs $S,T$ of $F$-linear subspaces in a field extension $L/F$ such that the linear span $ST$ of the set of products of elements of $S$ and of elements of $T$ has small dimension. Our central result is a linear analogue of Vosper's Theorem, which gives the structure of vector spaces $S, T$ in a prime extension $L$ of a finite field $F$ for which $\dim_FST =\dim_F S+\dim_F T-1,$ when $\dim_F S, \dim_F T\ge 2$ and $\dim_F ST\le [L:F]-2$., 33 pages

Using dynamical methods we give a new proof of the theorem saying that if $A,B,X$ are rational functions of degree at least two such that $A\circ X=X\circ B$ and $\mathbb C(B,X)=\mathbb C(z)$, then the Galois closure of the field extension $\mathbb C(z)/\mathbb C(X)$ has genus zero or one., An extended version

J. R. Sendra and C. Villarino are members of the research group ASYNACS (Ref. CCEE2011/R34)., In this extended abstract, we study the properties of ultraquadrics associated with automorphisms of the field K(\alpha )(t1, . . . , tn), defined by linear rational (with common denominator) or by polynomial (with polynomial inverse) coordinates. We conclude that ultraquadrics related to polynomial automorphisms can be characterized as varieties K−isomorphic to linear varieties, while ultraquadrics arising from projective automorphisms are isomorphic to the Segre embedding of a blowup of the projective space along an ideal and, in some general case, linearly isomorphic to a toric variety. This information helps us to compute a parametrization of some ultraquadrics., Ministerio de Ciencia e Innovación