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Generic polynomial
About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.
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TL;DR: In this paper, it was shown that there exists a linear representation such that the Artin L -function for p is equal to the L-function associated to f(z).
Abstract: Let f(z) be a cusp form of type (l,e) on Γ 0 (N) which is a common eigenfunction of all Hecke operators. For such f(z) , Deligne and Serre [1] proved that there exists a linear representation such that the Artin L -function for p is equal to the L -function associated to f(z) .
11 citations
TL;DR: The totally real algebraic number field F of degree 6 with Galois group A 6 and minimum discriminant is determined and is unique up to isomorphy, and is generated by a root of the polynomial t 6 over the rationals.
Abstract: The totally real algebraic number field F of degree 6 with Galois group A 6 and minimum discriminant is determined It is unique up to isomorphy, and is generated by a root of the polynomial t 6 — 24t 4 + 2lt 2 + 9t + 1 over the rationals We also give an integral basis and list the fundamental units and class number of F
11 citations
TL;DR: A Grunwald-Wang-type theorem whenever there is a generic extension for a group G over a field F and such that L superset, dbl equals F is found.
Abstract: We define the notion of a generic Galois extension with group G over a field F. Let R be a communtative ring of the form F[x1,..., xn](1/s) and let S be a Galois extension of R with group G. Then S/R is generic for G over F if the following holds. Assume K/L is a Galois extension of fields with group G and such that L ⊇ F. Then there is an F algebra map f:R → L such that K ≅ S [unk]RL. We construct generic Galois extensions for certain G and F. We show such extensions are related to Noether's problem and the Grunwald-Wang theorem. One consequence is a simple proof of known counter examples to Noether's problem. On the other hand, we have an elementary proof of a chunk of the Grunwald-Wang theorem, and in a more general context. In fact, we have a Grunwald-Wang-type theorem whenever there is a generic extension for a group G over a field F.
11 citations
TL;DR: In this article, the Galois group of the maximal unramified p-extension of an imaginary quadratic field is investigated for the case where the group is finite, and it is shown that the group can be generated by not more than two generators with two relations.
Abstract: The Galois group of the maximal unramified p-extension (p 2) of an imaginary quadratic field is investigated for the case where the group is finite. It is shown that the group can be generated by not more than two generators with two relations. One of the relations can be taken from the 3rd term of the Zassenhaus filtration of the free group, and the second, from the 2nd, 5th, or 7th term.
11 citations
Proceedings Article•
11 Oct 2012
TL;DR: A new efficient algorithm for building a Galois sub-hierarchy which runs in O(min{nm, n α }), where n is the number of objects or attributes, m is the size of the relation, and n α is the time required to perform matrix multiplication.
Abstract: Given a relation R ⊆ O × A on a set O of objects and a set A of attributes, the Galois sub-hierarchy (also called AOC-poset) is the partial order on the introducers of objects and attributes in the corresponding concept lattice. We present a new efficient algorithm for building a Galois sub-hierarchy which runs in O(min{nm, n α }), where n is the number of objects or attributes, m is the size of the relation, and n α is the time required to perform matrix multiplication (currently α = 2.376).
11 citations