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We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain A with 1 and a polynomial equation antn + ...+ a0 = 0 with coefficients ai in A, our problem is to find its roots in A.We show that when A = B[x] is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over B. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over A when A is F[x1, ..., xN] or F(x1,...
In this article we extend the algebraic theory of polynomial rings, formalized in Mizar [1], based on [2], [3]. After introducing constant and monic polynomials we present the canonical embedding of R into R[X] and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields F and irreducible polynomials p the field F[X]/ is isomorphic to the field of polynomials with degree smaller than the one of p.
Dans cet article, nous tentons de généraliser à d’autres situations l’isomorphisme de groupes topologiques qui existe entre le groupe et le groupe unitaire .Nous montrons que cet isomorphisme existe algébriquement en toute généralité : pour tout corps algébriquement clos et toute involution de les groupes et sont isomorphes. Nous donnons ensuite un exemple d’involution de qui n’est pas conjuguée, dans le groupe , à la conjugaison complexe et telle que soit topologiquement isomorphe...
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