Search

Let v be a Krull valuation of a field K with valuation ring R v and K 1 , K 2 be finite separable extensions of K which are linearly disjoint over K. Assume that the integral closure of R v in the composite field K 1 K 2 is a free R v -module. For a given pair of prolongations v 1 , v 2 of v to K 1 , K 2 respectively, it is shown that there exists a unique prolongation w of v to K 1 K 2 which extends both v 1 , v 2 . Moreover with S i as the integral closure of R v in K i , if the ring S 1 S 2 is integrally closed and the residue field of v is perfect, then f ( w / v ) = f ( v 1 / v ) f ( v 2 / v ) , where f ( v ′ / v ) stands for the degree of the residue field of a prolongation v ′ of v over the residue field of v. As an application, it is deduced that if K 1 , K 2 are algebraic number fields which are linearly disjoint over K = K 1 ∩ K 2 , then the number of prime ideals of the ring A K 1 K 2 of algebraic integers of K 1 K 2 lying over a given prime ideal ℘ of A K equals the product of the numbers of prime ideals of A K i lying over ℘ for i = 1 , 2 .

It is well known that if K 1 , K 2 are algebraic number fields with coprime discriminants, then the composite ring A K 1 A K 2 is integrally closed and K 1 , K 2 are linearly disjoint over the field of rationals, A K i being the ring of algebraic integers of K i . In an attempt to prove the converse of the above result, in this paper we prove that if K 1 , K 2 are finite separable extensions of a valued field ( K , v ) of arbitrary rank which are linearly disjoint over K = K 1 ∩ K 2 and if the integral closure S i of the valuation ring R v of v in K i is a free R v -module for i = 1 , 2 with S 1 S 2 integrally closed, then the discriminant of either S 1 / R v or of S 2 / R v is the unit ideal. We quickly deduce from this result that for algebraic number fields K 1 , K 2 linearly disjoint over K = K 1 ∩ K 2 for which A K 1 A K 2 is integrally closed, the relative discriminants of K 1 / K and K 2 / K must be coprime.

For an extension L/K of discrete valuation fields, let $$e_{L/K}$$ , $${\mathfrak {O}}_{L}$$ denote the ramification index and valuation ring of L/K respectively. Let K be a complete discrete valuation field and $$L_1/K$$ , $$L_2/K$$ be finite linearly disjoint extensions over K. We show that if $${\mathfrak {O}}_{L_1L_2} ={\mathfrak {O}}_{L_1}{\mathfrak {O}}_{L_2}$$ or $$\mathrm {gcd}(e_{L_1/K}, e_{L_2/K}) =1$$ , and one of the residue fields $$l_1/k,$$ $$l_2/k$$ is separable, then $$e_{L_1L_2/L_1} =e_{L_2/K}.$$ The analogous results for the residue degrees are also true.

We construct the counter-example for polynomial version of Sarnak's conjecture for minimal systems, which assets that the M\"obius function is linearly disjoint from subsequences along polynomials of deterministic sequences realized in minimal systems. Our example is in the class of Toeplitz systems, which are minimal., Comment: Final version. To appear in Adv. Math

We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end, we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.

We prove that the flow generated by any continuous interval map with zero topological entropy is minimally mean-attractable and minimally mean-L-stable. One of the consequences is that any oscillating sequence is linearly disjoint from all flows generated by all continuous interval maps with zero topological entropy. In particular, the Mobius function is linearly disjoint from all flows generated by all continuous interval maps with zero topological entropy (Sarnak's conjecture for continuous interval maps). Another consequence is a non-trivial example of a flow having discrete spectrum. We also define a log-uniform oscillating sequence and show a result in ergodic theory for comparison.

Let K be a number field, and let X → P K 1 be a degree p covering branched only at 0, 1, and ∞. If K is a field containing a primitive p -th root of unity then the covering of P 1 is Galois over K , and if p is congruent to 1 mod 6 , then there is an automorphism σ of X which cyclically permutes the branch points. Under these assumptions, we show that the Jacobian of both X and X / 〈 σ 〉 gain rank over infinitely many linearly disjoint cyclic degree p -extensions of K . We also show the existence of an infinite family of elliptic curves whose j -invariants are parametrized by a modular function on Γ 0 ( 3 ) and that gain rank over infinitely many cyclic degree 3-extensions of Q .

In this paper we study $C^*$-algebra version of Sarnak Conjecture for noncommutative toral automorphisms. Let $A_\Theta$ be a noncommutative torus and $\alpha_\Theta$ be the noncommutative toral automorphism arising from a matrix $S\in GL(d,\mathbb{Z})$. We show that if the Voiculescu-Brown entropy of $\alpha_{\Theta}$ is zero, then the sequence $\{\rho(\alpha_{\Theta}^nu)\}_{n\in \mathbb{Z}}$ is a sum of a nilsequence and a zero-density-sequence, where $u\in A_\Theta$ and $\rho$ is any state on $A_\Theta$. Then by a result of Green and Tao, this sequence is linearly disjoint from the M\"obius function., Comment: 29 pages

Let \begin{document}$ \mathbb{T} $\end{document} be the unit circle and \begin{document}$ \Gamma \backslash G $\end{document} the \begin{document}$ 3 $\end{document} -dimensional Heisenberg nilmanifold. We prove that a class of skew products on \begin{document}$ \mathbb{T} \times \Gamma \backslash G $\end{document} are distal, and that the Mobius function is linearly disjoint from these skew products. This verifies the Mobius Disjointness Conjecture of Sarnak.

We define anoscillating sequence, an important example of which is generated by the Möbius function in number theory. We also define aminimally mean attractable (MMA) flowand aminimally mean-L-stable (MMLS) flow. One of the main results is that any oscillating sequence is linearly disjoint from all MMA and MMLS flows. In particular, this confirms Sarnak’s conjecture for all MMA and MMLS flows. We provide several examples of flows that are MMA and MMLS. These examples include flows defined by all$p$-adic polynomials of integral coefficients, all$p$-adic rational maps with good reduction, all automorphisms of the$2$-torus with zero topological entropy, all diagonalizable affine maps of the$2$-torus with zero topological entropy, all Feigenbaum maps, and all orientation-preserving circle homeomorphisms.

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.

Let K be a composite field of a cyclotomic field k n of odd conductor n ≧ 3 or even one ≧8 with 4 | n and a totally real algebraic extension field F over the rationals Q and both fields k n and F are linearly disjoint over Q to each other. Then the purpose of this paper is to prove that such a relatively totally real extension field K over a cyclotomic field k n has no power integral basis. Each of the composite fields K is also a CM field over the maximal real subfield k n + ⋅ F of K. This result involves the previous work for K = k n ⋅ F of the Eisenstein field k n = k 3 and the maximal real subfields F = k p n + of prime power conductor p n with p ≧ 5 , and an analogue K = k n ⋅ F of cyclotomic fields k n = k 2 m ( m ≧ 3 ) with a totally real algebraic fields F of K = k 4 ⋅ F with a cyclic cubic field F except for k 4 ⋅ k 7 + and k 4 ⋅ k 3 2 + of conductors 28 and 36.

A theorem of Albert-Draxl states that if a tensor product of two quaternion division algebras $Q_1$, $Q_2$ over a field $F$ is not a division algebra, then there exists a separable quadratic extension of $F$ that embeds as a subfield in $Q_1$ and in $Q_2$. We establish a modified version of this result where the tensor product of quaternion algebras is replaced by the corestriction of a single quaternion algebra over a separable field extension. As a tool in the proof, we show that if the transfer of a nonsingular quadratic form $\varphi$ over a quadratic extension is isotropic for a linear functional $s$ such that $s(1)=0$, then $\varphi$ contains a nondegenerate subform defined over the base field.

Shimura and Taniyama proved that if $A$ is a potentially CM abelian variety over a number field $F$ with CM by a field $K$ linearly disjoint from F, then there is an algebraic Hecke character $\lambda_A$ of $K$ such that $L(A/F,s)=L(\lambda_A,s)$. We consider a certain converse to their result. Namely, let $A$ be a potentially CM abelian variety appearing as a factor of the Jacobian of a curve of the form $y^e=\gamma x^f+\delta$. Fix positive integers $a$ and $n$ such that $n/2 < a \leq n$. Under mild conditions on $e, f, \gamma, \delta$, we construct a Chow motive $M$, defined over $F=\mathbb{Q}(\gamma,\delta)$, such that $L(M/F,s)$ and $L(\lambda_A^a\bar{\lambda}_A^{n-a},s)$ have the same Euler factors outside finitely many primes., Comment: 20 pages

In this paper we characterize the monogenity of non-cyclic but abelian octic number fields $$L$$ over the rationals $${{{\varvec{Q}}}},$$ each of which is composed by a linearly disjoint cyclic quartic field of odd prime conductor $$\ell $$ and a quadratic field of prime discriminant $$p^{*}.$$ In the case of each odd conductor $$\ell |p^{*}|,$$ the linear Diophantine equation with unit coefficients in a specified quadratic subfield of $$L$$ is applied to determine the monogenity of the octic field $$L.$$ In the case of each even conductor, among seven octic fields $$L$$ of conductor $$5|2^{*}|,$$ it is shown that the unknown four maximal imaginary subfields $$L$$ of the $$40$$ th cyclotomic field are non-monogenic, in which the non-monogenity of two fields are proved by the evaluation of the absolute norm of a partial factor $${\xi }-{\xi }^{{\rho }}$$ of the different $$\mathfrak {d}({\xi })$$ with a suitable Galois action $${\rho }$$ for any integer $${\xi }$$ in $$L$$ .

For all positive integers $\ell$, we prove non-trivial bounds for the $\ell$-torsion in the class group of $K$, which hold for almost all number fields $K$ in certain families of cyclic extensions of arbitrarily large degree. In particular, such bounds hold for almost all cyclic degree-$p$-extensions of $F$, where $F$ is an arbitrary number field and $p$ is any prime for which $F$ and the $p$-th cyclotomic field are linearly disjoint. Along the way, we prove precise asymptotic counting results for the fields of bounded discriminant in our families with prescribed splitting behavior at finitely many primes., Comment: Dedicated to Professor Robert F. Tichy on the occasion of his 60th birthday; 22 pages

An elementary proof using matrix theory is given for the following criterion: if F/K and L/K are field extensions, with F and L both contained in a common extension field, then F and L are linearly disjoint over K if (and only if) some K-vector space basis of F is linearly independent over L. The material in this note could serve as enrichment material for the unit on fields in a first course on abstract algebra.

Let$K$be a finitely generated extension of$\mathbb {Q}$. We consider the family of$\ell $-adic representations ($\ell $varies through the set of all prime numbers) of the absolute Galois group of$K$, attached to$\ell $-adic cohomology of a separated scheme of finite type over$K$. We prove that the fields cut out from the algebraic closure of$K$by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question of Serre.

During the Arizona Winter School 2008 (held in Tucson, AZ) we worked on the following problems: a) (Expanding a remark by S. Lang). Define $E_0 = \overline{\mathbb{Q}}$ Inductively, for $n \geq 1$, define $E_n$ as the algebraic closure of the field generated over $E_{n-1}$ by the numbers $\exp(x)=e^x$, where $x$ ranges over $E_{n-1}$. Let $E$ be the union of $E_n$, $n \geq 0$. Show that Schanuel's Conjecture implies that the numbers $\pi, \log \pi, \log \log \pi, \log \log \log \pi, \ldots $ are algebraically independent over $E$. b) Try to get a (conjectural) generalization involving the field $L$ defined as follows. Define $L_0 = \overline{\mathbb{Q}}$. Inductively, for $n \geq 1$, define $L_n$ as the algebraic closure of the field generated over $L_{n-1}$ by the numbers $y$, where $y$ ranges over the set of complex numbers such that $e^y\in L_{n-1}$. Let $L$ be the union of $L_n$, $n \geq 0$. We were able to prove that Schanuel's Conjecture implies $E$ and $L$ are linearly disjoint over $\overline{\mathbb{Q}}$., Comment: 8 pages summarizing the results obtained in this project during the AWS08 http://swc.math.arizona.edu/aws/08/08WaldschmidtOutline.pdf

We prove that the M\"{o}bius function is linearly disjoint from an analytic skew product on the $2$-torus. These flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points. The previous cases for which such disjointness has been proved are all regular. We also establish the linear disjointness of M\"{o}bius from various distal homogeneous flows., Comment: 42 pages

In his previous papers (J. reine angew. Math. 544 (2002), 91--110; math.AG/0103203) the author introduced a certain explicit construction of superelliptic jacobians, whose endomorphism ring is the ring of integers in the $p$th cyclotomic field. (Here $p$ is an odd prime.) In the present paper we discuss when these jacobians are mutually non-isogenous. (The case of hyperelliptic jacobians was treated in author's e-print math.NT/0301173 .), LaTeX2e, 16 pages. We extend the main result to a broader class of superelliptic jacobians related to polynomials with doubly transitive Galois groups

For a complete discrete valuation field K of characteristic 0 with residue field of characteristic p > 0, consider the problem of embedding a given cyclic extension M/K of degree p into a cyclic extension of degree pn for various n. Let c(M/K) be equal to the maximal n for which this embedding problem has a solution. In this paper we consider relations between c(M/K) and c(LM/L), where L/K is a given extension linearly disjoint with M/K. Bibliography: 5 titles.

Let M be a field of positive characteristic p > 0 such that C, the closure of a finite field in M, has an extension of degree p. Let L be a field finitely generated over C and such that M and L are linearly disjoint over C. Then Hilbert’s tenth problem is not decidable over ML. Bibliography: 41 titles.

We investigate the issues of Diophantine definability over the non-finitely generated version of non-degenerate modules contained in the infinite algebraic extensions of the rational numbers. In particular, we show the following. Let k be a number field and let K inf be a normal algebraic, possibly infinite, extension of k such that k has a normal extension L linearly disjoint from K inf over k. Assume L is totally real and K inf is totally complex. Let M inf be a non-degenerate O k -module, possibly non-finitely generated and contained in O Kinf . Then M inf contains a submodule M¯ inf such that M inf /M¯ inf is torsion and O k has a Diophantine definition over M¯ inf .

Text Let L 1 and L 2 be finite separable extensions of a global field K , and let E i be the Galois closure of L i over K for i = 1 , 2 . We establish a local-global principle for the product of norms from L 1 and L 2 (so-called multinorm principle ) provided that the extensions E 1 and E 2 are linearly disjoint over K . Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=6fwHt_k5dVQ .

Resume Letkbe a number field,Okits ring of integers,Γa nonabelian metacyclic group of orderlq, wherel,qare prime numbers. Assume thatkand thelqth cyclotomic field of Q are linearly disjoint over Q . Let M be a maximalOk-order ink[1] containingOk[Γ], Cl ( M ) its class group. We determine the set of elements of Cl ( M ) which are realizable by some tamely ramified extensions with Galois groups isomorphic toΓusing some Stickelberger elements, and we prove that it is a subgroup of Cl ( M ).

For valued fields $K$ of rank higher than 1, we describe how elements in the henselization $K^h$ of $K$ can be approximated from within $K$; our result is a handy generalization of the well-known fact that in rank 1, all of these elements lie in the completion of $K$. We apply the result to show that if an element $z$ algebraic over $K$ can be approximated from within $K$ in the same way as an element in $K^h$, then $K(z)$ is not linearly disjoint from $K^h$ over $K$., Comment: 13 pages

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, \rho_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of \rho_E for a "random" elliptic curve E over k. In particular, if k\neq Q is linearly disjoint from the cyclotomic extension of Q, then \rho_E will be surjective for "most" elliptic curves over k., Comment: 14 pages

We prove that if K is a subextension of \(\tilde{\Bbb Q} / \Bbb Q\) which is PAC over \(\Bbb Q \), then the extension \(\tilde{\Bbb Q} / \Bbb Q\)splits through K, i.e., there is a field \(L\subseteq \tilde{\Bbb Q}\) linearly disjoint from K over \(\Bbb Q \) such that \(KL=\tilde{\Bbb Q}\).

We describe a method for proving the existence of a dierential field over which two given dierential subfields are linearly disjoint. In the armative case, they are linear disjoint over their intersection, which can be computed using characteristic sets. Our method is based on the correspon- dence between field extensions and ideals.

Let L L be a function field in n > 0 n > 0 variables over a field K K of characteristic p ≠ 0 p \ne 0 . An intermediate field S S is maximal separable if S S is separable over K K and every subfield of L L which properly contains S S is inseparable over K K . This paper examines when [ L : S ] | S [L:S]|S is maximal separable is bounded. The main result states that this set is bounded if and only if there is an integer c c such that any intermediate field L 1 {L_1} over which L L is purely inseparable and [ L : L 1 ] > p c [L:{L_1}] > {p^c} must be separable over K K . Examples are also given where the above bound is p n + 1 {p^{n + 1}} for any n ⩾ 1 n \geqslant 1 .

Baur a défini la notion d'extension séparée de corps valués et montré que toute extension d'un corps maximal est séparée. Nous prouvons que, si (K, υ) est henselien et de caractéristique résiduelle nulle, alors (K, υ) ⊂ (L, w) est séparée ssi L est linéairement disjoint sur K de toute extension immédiate de K.Separated and immediate extensions of valued fields. The notion of separated extension of valued fields was introduced by Baur. He showed that extensions of maximal fields are separated. We prove that, when (K, υ) is Henselian with residual characteristic 0, then (K, υ) ⊂ (L, w) is separated iff L is linearly disjoint over K from each immediate extension of K.

Let F F be a skew field and C C a central subfield, then the free F F -field on X X centralizing C C is denoted by F C ( X ) {F_C}(X) . The object is to prove the following theorem. Let F F be a skew field with a central subfield C C , let E E be a subfield of F F and put k = E ∩ C k = E \cap C ; then there is a natural embedding of E k ( X ) {E_k}(X) in F C ( X ) {F_C}(X) if and only if E E and C C are linearly disjoint over k k . This result replaces the erroneous Theorem 6.3.6 on p. 148 of the author’s Skew field constructions, a counterexample to the latter (due to G. M. Bergman) is also described. The paper also includes an improved form of the specialization lemma (1.c.).

PROOF. In view of the above remarks we may assume k/k' is purely inseparable. Let p be the characteristic. Then k/k' is a p-tower in which each step is of degree p and is inseparable. We may further assume that (k: k') = p because a subfield of a separably generated field is also separably generated. (This is an immediate consequence of MacLane's criterion that k is separably generated over ko if and only if k is linearly disjoint from k'lp over ko.) Let x be a separating variable for k over ko so that we may write k=ko(x, y) where y is separable over ko(x).. Then we also have k=ko(x, yP). We see that ko(xP, yP)Ck' and in fact we must have k ko(XP yP) because

Throughout this note L/K denotes a purely inseparable field extension of characteristic p and nonzero exponent. In [5, p. 745], Rygg proves that when L/K has bounded exponent, then a subset M of L is a relative ?-base of L/K if and only if M is a minimal generating set of L/K. The purpose of this note is to answer the following question : If every relative ?-base of L/K is a minimal generating set, then must L/K he of bounded exponent? The answer is known to be yes when K and 2>* are linearly disjoint, * = 1, 2, • • • , see [l]. We give two examples for which the answer is no: One in which the maximal perfect subfield of L is contained in K, and the other in which it is not. The following lemmas are needed for our examples. An intermediate field V of L/K is called proper if KCLL'EL.

If G is a finite group and k is a field, then G is k-admissible if there exists a G-Galois extension L/k such that L is a maximal subfield of a k-division algebra. We prove that PSL(2, 7) is k-admissible for any number field which either fails to contain −1 or which has two primes lying over the dyadic prime. In addition, PSL(2, 11) is shown to be admissible over Q or any number field k with at least two extensions of the dyadic prime. Indeed, there exist infinitely many linearly disjoint admissible extensions for these groups.

Let k be a number field and O k its ring of integers. Let Γ be the alternating group A 4 . Let M be a maximal O k -order in k [ Γ ] containing O k [ Γ ] and C l( M ) its class group. We denote by R ( M ) the set of realizable classes, that is the set of classes c∈ C l( M ) such that there exists a Galois extension N / k at most tamely ramified, with Galois group isomorphic to Γ , for which the class of M ⊗ O k [Γ] O N is equal to c , where O N is the ring of integers of N . In this article we determine R ( M ) and we prove that it is a subgroup of C l( M ) provided that k and the 3rd cyclotomic field of Q are linearly disjoint, and the class number of k is odd.

Let L = L 1 ( x 1 ) = L 2 ( x 2 ) ⊃ K L = {L_1}({x_1}) = {L_2}({x_2}) \supset K where x i {x_i} is transcendental over L i {L_i} , and L i {L_i} is a finitely generated transcendence degree 1 extension of K K , i = 1 , 2 i = 1,2 . If the genus of L 1 / K = 0 {L_1}/K = 0 , then L 1 {L_1} and L 2 {L_2} are K K -isomorphic. If the genus of L 1 / K > 0 {L_1}/K > 0 , then L 1 = L 2 {L_1} = {L_2} and moreover L 1 {L_1} is invariant under all automorphisms of L / K L/K . A criterion is also established for a subfield of a ruled field L L to be ruled.

The purpose of this paper is to give a brief general account of the completely reducible finite-dimensional representations of a locally finite group G over a given algebraically closed field K. Theorem 1 shows that all such representations of G can be brought down to the algebraic closure F in K of the prime field of K. This reduces all further considerations in this account to countable groups. Theorem 2 characterizes the existence of a faithful completely reducible representation of G of degree n over K in terms of the existence of such representations for appropriate finite subgroups of G. Throughout the paper, G denotes a locally finite group, K denotes an arbitrary algebraically closed field and F denotes the algebraic closure in K of the prime field of K. F denotes an w-dimensional vector space over K. An F-form of V is an -F-subspace W of V such that W and K are linearly disjoint over K and Vis the X-span of W. (Equivalently, an F-form of V is the i^-span of a basis of V.) Il A is an Falgebra, AK denotes the algebra A®pK.