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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Resolvent quadratic ： ウィキペディア英語版 Resolvent (Galois theory) In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group ''G'' is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a rational root if and only if the Galois group of ''p'' is included in ''G''. More exactly, if the Galois group is included in ''G'', then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are * $X^2-\backslash Delta$ where $\backslash Delta$ is the discriminant, which is a resolvent for the alternating group. In the case of a cubic equation, this resolvent is sometimes called the quadratic resolvent; its roots appear explicitly in the formulas for the roots of a cubic equation. * The cubic resolvent of a quartic equation, which is a resolvent for the dihedral group of 8 elements. * The Cayley resolvent is a resolvent for the maximal resoluble Galois group in degree five. It is a polynomial of degree 6. These three resolvents have the property of being ''always separable'', which means that, if they have a multiple root, then the polynomial ''p'' is not irreducible. It is not known if there is an always separable resolvent for every group of permutations. For every equation the roots may be expressed in terms of radicals and of a root of a resolvent for a resoluble group, because, the Galois group of the equation over the field generated by this root is resoluble. == Definition == Let be a positive integer, which will be the degree of the equation that we will consider, and $(X\_1,\; \backslash ldots,\; X\_n)$ an ordered list of indeterminates. This defines the ''generic polynomial'' of degree :$F(X)=X^n+\backslash sum\_^n\; (-1)^i\; E\_i\; X^\; =\; \backslash prod\_^n\; (X-X\_i),$ where is the ''i''th elementary symmetric polynomial. The symmetric group acts on the by permuting them, and this induces an action on the polynomials in the . The stabilizer of a given polynomial under this action is generally trivial, but some polynomials have a bigger stabilizer. For example, the stabilizer of an elementary symmetric polynomial is the whole group . If the stabilizer is non-trivial, the polynomial is fixed by some non-trivial subgroup ; it is said an ''invariant'' of . Conversely, given a subgroup of , an invariant of is a resolvent invariant for if it is not an invariant of any bigger subgroup of .〔http://www.alexhealy.net/papers/math250a.pdf〕 Finding invariants for a given subgroup of is relatively easy; one can sum the orbit of a monomial under the action of . However it may occur that the resulting polynomial is an invariant for a larger group. For example, let us consider the case of the subgroup of of order 4, consisting of , , and the identity (for the notation, see Permutation group). The monomial $X\_1\; X\_2$ gives the invariant $2(X\_1\; X\_2\; +\; X\_3\; X\_4)$. It is not a resolvent invariant for , as being invariant by . In fact, it is a resolvent invariant for the dihedral subgroup $\backslash langle\; (12),\; (1324)\; \backslash rangle$, and is used to define the cubic resolvent of the quartic equation. If is a resolvent invariant for a group of index , then its orbit under has order . Let $P\_1,\; \backslash ldots,\; P\_m$ be the elements of this orbit. Then the polynomial :$R\_G=\backslash prod\_^m\; (Y-P\_i)$ is invariant under . Thus, when expanded, its coefficients are polynomials in the that are invariant under the action of the symmetry group and thus may be expressed as polynomials in the elementary symmetric polynomials. In other words, is an irreducible polynomial in whose coefficients are polynomial in the coefficients of . Having the resolvent invariant as a root, it is called a resolvent (sometimes resolvent equation). Let us consider now an irreducible polynomial :$f(X)=X^n+\backslash sum\_^n\; a\_i\; X^\; =\; \backslash prod\_^n\; (X-x\_i),$ with coefficients in a given field (typically the field of rationals) and roots in an algebraically closed field extension. Substituting the by the and the coefficients of by those of in what precedes, we get a polynomial $R\_G^(Y)$, also called ''resolvent'' or ''specialized resolvent'' in case of ambiguity). If the Galois group of is contained in , the specialization of the resolvent invariant is invariant by and is thus a root of $R\_G^(Y)$ that belongs to (is rational on ). Conversely, if $R\_G^(Y)$ has a rational root, which is not a multiple root, the Galois group of is contained in . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Resolvent (Galois theory)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.