INVARIANTS OF NUMBER FIELDS RELATED TO CENTRAL EMBEDDING PROBLEMS

Every central embedding problem over a number field becomes solvable after enlarging its kernel in a certain way. We show that these enlargements can be arranged in a universal way.

where X m is the map induced by Xm on cohomology see Hoechsmann, [I]).represented by a Galols co-cycle all of whose values are roots of unity, the algebra class ((c))splits and only if it splits locally at all places above p and and m this is the case if it splits at and (kv ):kv( m) E 0 rood pm for all v above p; P P (see classels [2], p. 191, 10.5 ff).It is clearly possible to find a smallest integer d d(k,p) depending only on k and p such that ((c)) splits with 2m + d.
For m instance, for k Q we can take d d(Q,p) 0 for all p.The p-exponent of E is the m smallest natural number n ) m such that the induced embedding problem E has a n solution which is ramified only at p and The smallest intege,r s )0 such that the p-exponent of every E is 2m + s is called the strong p-exponent of k (if it m exists). ( If p does not divide the class number of Q(Bp)then the strong p-exponent of every cyclotomlc field k Q (pl) exists and is equal to its (usual) p-exponent.
This can be shown as follows: Let E be a central embedding problem for G k. Then for m t t(k,p) the induced embedding problem E2m + t is solvable.The assumption implies that p does not divide the class number of 0(Bpl)for every (see lwasawa [3]).
Therefore the Galols theoretic obstruction to the existence of a solution which is unramlfled outside p and as described in Neuklrch [4], (8.1), is trivial.
The p-adlc Leopoldt conjecture for k implies that H2(Gk(P),Q/Z) 0, where Gk(p) is the Galols group of the maximal p-extenslon kP/k which is unramlfled outside p and This shows that every central embedding problem E for Gk(p) has finite p- ro exponent, (see Opolka [5], (5.2)).Does this imply that the strong p-exponent of k is finite?
If so, how is it related to the usual p-exponent of k? Conversely, if the strong p-exponent of k is finite then H2(Gk(P),Q/Z) 0 and the p-adlc Leopoldt conjecture holds for k.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning prove (I), choose for any natural number m m an isomorphism m The exponent of E is the smallest natural number n > m such that the embedding problem E which is m n obtained from E by considering the co-cycle c:G x G z/pm+ Z/p n is solvable.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation