ترغب بنشر مسار تعليمي؟ اضغط هنا

Involutions on Zilber fields

151   0   0.0 ( 0 )
 نشر من قبل Vincenzo Mantova
 تاريخ النشر 2011
  مجال البحث
والبحث باللغة English
 تأليف Vincenzo Mantova




اسأل ChatGPT حول البحث

After recalling the definition of Zilber fields, and the main conjecture behind them, we prove that Zilber fields of cardinality up to the continuum have involutions, i.e., automorphisms of order two analogous to complex conjugation on (C,exp). Moreover, we also prove that for continuum cardinality there is an involution whose fixed field, as a real closed field, is isomorphic to the field of real numbers, and such that the kernel is exactly 2{pi}iZ, answering a question of Zilber, Kirby, Macintyre and Onshuus. The proof is obtained with an explicit construction of a Zilber field with the required properties. As further applications of this technique, we also classify the exponential subfields of Zilber fields, and we produce some exponential fields with involutions such that the exponential function is order-preserving, or even continuous, and all of the axioms of Zilber fields are satisfied except for the strong exponential-algebraic closure, which gets replaced by some weaker axioms.



قيم البحث

اقرأ أيضاً

We consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augment ed with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.
We revisit the geometry of involutions in groups of finite Morley rank. Our approach unifies and generalises numerous results, both old and recent, that have exploited this geometry; though in fact, we prove much more. We also conjecture that this pa th leads to a new identification theorem for $operatorname{PGL}_2(mathbb{K})$.
65 - Dabin Zheng , Mu Yuan , Nian Li 2018
An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications such as cryptography and coding theory. As far as we know, there are no t many involutions, and there isnt a general way to construct involutions over finite fields. This paper gives a necessary and sufficient condition for the polynomials of the form $x^rh(x^s)in bF_q[x]$ to be involutions over the finite field~$bF_q$, where $rgeq 1$ and $s,|, (q-1)$. By using this criterion we propose a general method to construct involutions of the form $x^rh(x^s)$ over $bF_q$ from given involutions over the corresponding subgroup of $bF_q^*$. Then, many classes of explicit involutions of the form $x^rh(x^s)$ over $bF_q$ are obtained.
Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. Let $K$ be a regular field which is not generically stable and let $p$ be its global generic type. We observe that if $K$ has a finite extension $L$ of degree $n$, then $p^{(n)}$ has unbounded orbit under the action of the multiplicative group of $L$. Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique non-trivial conjugacy class, and we notice that a classical group with one non-trivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then we construct a group of cardinality $omega_1$ with only one non-trivial conjugacy class and such that the centralizers of all non-trivial elements are countable.
We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F. We show that the behavior of this group, even when A is large, depends essentially on the roots of unity in F. For almost all completions of the theory of pseudo-finite fields we show that algebraic closure agrees with definable closure, as soon as A contains the relative algebraic closure of the prime field.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا