Finite Field Extension Research Articles

On the Brumer--Stark conjecture

Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer–Stark conjecture states that the Stickelberger element $\Theta_{S,T}^{H/F}$ annihilates the $T$-smoothed class group $\mathrm{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with ${\bf Z}[1/2]$. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of $\mathrm{Cl}^T(H) \otimes {\bf Z}[1/2]$ in terms of Stickelberger elements. We also show that this stronger result implies Rubin's higher rank version of the Brumer--Stark conjecture, again away from 2. Our technique is a generalization of Ribet's method, building upon on our earlier work on the Gross--Stark conjecture. Here we work with group ring valued Hilbert modular forms as introduced by Wiles. A key aspect of our approach is the construction of congruences between cusp forms and Eisenstein series that are stronger than usually expected, arising as shadows of the trivial zeroes of $p$-adic $L$-functions. These stronger congruences are essential to proving that the cohomology classes we construct are unramified at $p$.

Decomposition of idempotent 2-cocycles

Let $L$ be a finite Galois field extension of $K$ with Galois group $G$. We decompose any idempotent 2-cocycle $f$ using finite sequences of descending two-sided ideals of the corresponding weak crossed product algebra $A_f$. We specialize the results in

The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields

<abstract><p>Let $ q $ be an even prime power and let $ \mathbb{F}_{q} $ be the finite field of $ q $ elements. Let $ f $ be a nonzero polynomial over $ \mathbb{F}_{q^2} $ of the form $ f = a_{1}x_{1}^{m_{1}}+\dots+a_{s}x_{s}^{m_{s}}+y_{1}y_{2}+\dots+y_{n-1}y_{n}+y_{n-2t-1}^{2}+\dots +y_{n-3}^2+y_{n-1}^{2}$ $+b_{t}y_{n-2t}^{2}+ $ $ \dots +b_{1}y_{n-2}^{2}+b_{0}y_{n}^{2} $, where $ a_{i}, b_j\in \mathbb{F}_{q^{2}}^{*}, $ $ m_i\ne 1, $ $ (m_{i}, m_{k}) = 1, $ $ i\ne k, $ $ m_{i}|(q+1), $ $ m_{i}\in \mathbb{Z}^{+}, $ $ 2|n $, $ n > 2 $, $ 0\leq t\leq \frac{n}{2}-2 $, $ {\mathrm{Tr}}_{\mathbb{F}_{q^2}/\mathbb{F}_{2}}(b_{j}) = 1 $ for $ i, k = 1, \dots, s $ and $ j = 0, 1, \dots, t $. For each $ b \in \mathbb{F}_{q^2} $, let $ N_{q^2}(f = b) $ denote the number of $ \mathbb{F}_{q^2} $-rational points on the affine hypersurface $ f = b $. In this paper, we obtain the formula of $ N_{q^2}(f = b) $ by using the Jacobi sums, Gauss sums and the results of quadratic form in finite fields.</p></abstract>

Ostrowski quotients for finite extensions of number fields

For $L/K$ a finite Galois extension of number fields, the relative P\'olya group $\Po(L/K)$ coincides with the group of strongly ambiguous ideal classes in $L/K$. In this paper, using a well known exact sequence related to $\Po(L/K)$, in the works of Brumer-Rosen and Zantema, we find short proofs for some classical results in the literatur. Then we define the ``Ostrowski quotient'' $\Ost(L/K)$ as the cokernel of the capitulation map into $\Po(L/K)$, and generalize some known results for $\Po(L/\mathbb{Q})$ to $\Ost(L/K)$.

Bazı Dembowski-Ostrom Polinomlarının Planaritesi Üzerine

Planar mappings, defined by Dembowski and Ostrom, are identified as a means to construct projective planes. Then, many important applications of planar mappings appear in different fields such as cryptography and coding theory. In this paper, we provide sufficient and necessary conditions for the planarity of certain Dembowski-Ostrom polynomials over the finite field extension of degree three with odd characteristic. In particular, we completely determine the coefficients of the given Dembowski-Ostrom polynomials to be planar.

Norm Principle for Even K-groups of Number Fields

We investigate the norm maps of algebraic even K-groups of finite extensions of number fields. Namely, we show that they are surjective in most situations. In the event that they are not surjective, we give a criterion in determining when an element in the even K-group of the base field comes from a norm of an element from the even K-groups of the extension field. This latter criterion is only reliant on the real primes of the base field.

Éz fields

Let K be a field. The étale open topology on the K-points V(K) of a K-variety V was introduced in [23]. The étale open topology is non-discrete if and only if K is large. If K is separably, real, p-adically closed then the étale open topology agrees with the Zariski, order, valuation topology, respectively. We show that existentially definable sets in perfect large fields behave well with respect to this topology: such sets are finite unions of étale open subsets of Zariski closed sets. This implies that existentially definable sets in arbitrary perfect large fields enjoy some of the well-known topological properties of definable sets in algebraically, real, and p-adically closed fields. We introduce and study the class of éz fields: K is éz if K is large and every definable set is a finite union of étale open subsets of Zariski closed sets. This should be seen as a generalized notion of model completeness for large fields. Algebraically closed, real closed, p-adically closed, and bounded PAC fields are éz. (In particular pseudofinite fields and infinite algebraic extensions of finite fields are éz.) We develop the basics of a theory of definable sets in éz fields. This gives a uniform approach to the theory of definable sets across all characteristic zero local fields and a new topological theory of definable sets in bounded PAC fields. We also show that some prominent examples of possibly non-model complete model-theoretically tame fields (characteristic zero t-Henselian fields and Frobenius fields) are éz.

Pathologies du groupe des classes de R-équivalence d’un groupe algébrique linéaire

Let \(k_0\) be a field of characteristic \(p>0\) and \(k=k_0(t)\), where t is transcendental over \(k_0\). We give an example of a smooth connected unipotent k-group G such that G(F)/R is non-commutative for some finite separable field extension F containing k.

The relative class number one problem for function fields, I

We reduce the classification of finite extensions of function fields (of curves over finite fields) with the same class number to a finite computation; complete this computation in all cases except when both curves have base field mathbb {F}_2 and genus >1; and give a conjectural answer in the remaining cases. The conjecture will be resolved in subsequent papers.

A Tale of Two Cyclicities: Counting in Finite Cyclic Groups and Finite Fields

We study several combinatorial identities that arise naturally from the structure of finite cyclic groups and finite extensions of finite fields. We are particularly interested in interconnections between the identities.

Higher Stickelberger ideals and even 𝐾-groups

We use the analogy between class groups and even K K -groups of the ring of integers of a number field and “Higher Stickelberger” ideals within K K -theory to prove an index formula for these ideals in a finite abelian extension of real number fields, which is similar to the classic Stickelberger ideal index formula proved by Iwasawa.

Orthogonality of invariant vectors

Let G be a finite group with given subgroups H and K. Let π be an irreducible complex representation of G such that its space of H-invariant vectors as well as the space of K-invariant vectors are both one dimensional. Let vH (resp. vK) denote an H-invariant (resp. K-invariant) vector of unit norm in a given G-invariant inner product 〈,〉π on π. Our interest is in computing the square of the absolute value of 〈vH,vK〉π. This is the correlation constant c(π;H,K) defined by Gross [3]. In this paper, we study this question for G=GL2(Fq), where Fq is the finite field of q=pm elements of odd characteristic p, H is its split torus and K is a non-split torus. The first main theorem of this paper gives an explicit formula for |〈vH,vK〉π|2 modulo p. The key idea here is to analyse the mod p reduction of π. The second main theorem relates the behaviour of 〈vH,vK〉π under Shintani base change and gives a sufficient condition for 〈vH,vK〉π to vanish for an irreducible representation π=BC(τ) of PGL2(E), in terms of the epsilon factor of the base changing representation τ of PGL2(F), where E/F is a finite extension of finite fields.

Algebraic spaces that become schematic after ground field extension

AbstractWe construct examples of non‐schematic algebraic spaces that become schemes after finite ground field extensions.

Catenarian FCP ring extensions

If R⊆S is a ring extension of commutative unital rings, the poset [R,S] of R-subalgebras of S is called catenarian if it verifies the Jordan–Hölder property. This property has already been studied by Dobbs and Shapiro for finite extensions of fields. We investigate this property for arbitrary ring extensions, showing that many types of extensions are catenarian. We reduce the characterization of catenarian extensions to the case of field extensions, an unsolved question at this time.

On [formula omitted]-Hopf-Galois structures

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 Zn and Z2. 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 Zn⋊ϕZ2 when n is odd with radical of n being a Burnside number. As an application we also find the corresponding number of skew braces.

On generalized strong complete mappings and mutually orthogonal Latin squares

We present an application of generalized strong complete mappings to construction of a family of mutually orthogonal Latin squares. We also determine a cycle structure of such mapping which form a complete family of MOLS. Many constructions of generalized strong complete mappings over an extension of finite field are provided.

Field Extensions Defined by Power Compositional Polynomials

Suppose $L/F$ is a finite extension of fields of characteristic 0. We study when $L$ can be defined by an irreducible power compositional polynomial; that is, a polynomial of the form $f(x^k)\in F[x]$. If $L$ contains the $k$th roots of unity, then Kummer theory provides an answer. In this case, $f(x^k)$ can be constructed explicitly using automorphisms of $L/F$. If $L$ does not contain the $k$th roots of unity, we use a result of Kang to give a complete answer when $k=3$ and $[L:F]=6$, and we construct a polynomial $x^6+ax^3+b$ defining $L/F$. As an application, we give a simple method for determining the Galois group of an irreducible polynomial $x^6+ax^3+b\in F[x]$.

Multiplicities in Selmer groups and root numbers of Artin twists

Let K/F be a finite Galois extension of number fields and let σ be an absolutely irreducible, self-dual, complex valued representation of Gal(K/F). Let p be an odd prime and consider two elliptic curves E1,E2 defined over Q with good, ordinary reduction at primes above p and equivalent mod-p Galois representations. In this article, we study the variation of the parity of the multiplicities of σ in the representation space associated to the p∞-Selmer groups of E1 and E2 over K. We also compare the root numbers for the twists of E1 and E2 over F by σ and show that the p-parity conjecture holds for the twist of E1/F by σ if and only if it holds for the twist of E2/F by σ. We also express Mazur-Rubin-Nekovář's arithmetic local constants in terms of certain local Iwasawa invariants.

Conjugation of semisimple subgroups over real number fields of bounded degree

Let G G be a linear algebraic group over a field k k of characteristic 0. We show that any two connected semisimple k k -subgroups of G G that are conjugate over an algebraic closure of k k are actually conjugate over a finite field extension of k k of degree bounded independently of the subgroups. Moreover, if k k is a real number field, we show that any two connected semisimple k k -subgroups of G G that are conjugate over the field of real numbers R {\mathbb {R}} are actually conjugate over a finite real extension of k k of degree bounded independently of the subgroups.

How far is an extension of p-adic fields from having a normal integral basis?

Let L/K be a finite Galois extension of p-adic fields with group G. It is well-known that OL contains a free OK[G]-submodule of finite index. We study the minimal index of such a free submodule, and determine it exactly in several cases, including for any cyclic extension of degree p of p-adic fields.