Stickelberger's discriminant relation
WebJul 24, 2024 · The Eigenvalue Theorem shows that solving a zero-dimensional polynomial system can be recast as an eigenvalue problem. This paper explores the relation between the Eigenvalue Theorem and the work of Ludwig Stickelberger (1850-1936). WebA classical result of Stickelberger (1897) [33] determines the parity of the number of irreducible factors of a squarefree polynomial in terms of the quadratic character of its discriminant. This was taken up by Dalen (1955) [10], and Swan (1962) [34] provides a simple formula for the discriminant of a trinomial. See also Golomb
Stickelberger's discriminant relation
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Webde 9 est la valeur absolue du discriminant d de L. Comme d a le signe de (- l)r2, on retrouve le résultat de Stickelberger publié en 1897 dans les actes du premier congrès international([9]), résultat dont une démonstration très simple a été donnée par Schur en 1928 ([7]) : 1.2 Corollaire. Le discriminant d'un corps de nombres est ... WebClassical proofs of Stickelberger’s congruences make use of the fact that any odd discriminant ideal d L/K is canonically associated with the discrimi-nant of a quadratic extension of K, unramified at 2. This essential reduction is summarized in the following proposition (see [Ma, § 3]). Proposition 3.
Webthe (Galois-module action of) the so-called Stickelberger ideal. Under some plausible number-theoretical hypothesis, our approach provides a ... it provides explicit class relations between an ideal and its Galois conjugates. ... discriminant ∆ K ofK ... Webnew proof of Stickelberger’s theorem even in the case of the ring of integers of a number eld. Moreover, our proof introduces a new invariant of a ring of rank nequipped with a …
WebUsing Stickelberger’s theorem (later rediscovered by Swan) one can determine the parity of the number of irreducible factors of a given square-free univariate polynomial over a finite field. This is done by examining either the discriminant of the given polynomial or the discriminant of its lift to the integers. Webon Stickelberger′s congruences for the absolute norms of relative discriminants of number fields, by using classical arguments of class field theory. 1. Introduction Let L/K be a finite extension of number fields. Denote by d L/K the relative discriminant of L/K and by c the number of complex infinite places of L which lie above a real ...
WebThe theorem of Stickelberger-Voronoi (cf. N. G. Chebotarev, Foundations of Galois Theory [in Russian], Vol. 2 (1937), p. 75) is extended to two unramified prime numbers in an algebraic number field. The proof is based on the following result: let K/k be a Galois extension of an arbitrary field {ik},Clar k≠2; the discriminant of the extension is not a …
WebAs you say, by Stickelberger's Theorem, the discriminant of any number field is $0$ or $1$ modulo $4$. Conversely, if $d \equiv 1 \pmod {4}$ is squarefree, then the discriminant of $\mathbb {Q} (\sqrt {d})$ is $d$. An integral basis for the ring of integers in this case is $\ {1$, $\frac {1+\sqrt {d}} {2}\}$. every mind matters insomniaWebApr 1, 1985 · Indeed the Stickelberger ideal S of k is closely related with the relative class number h- of k (cf. [4, 5, 8, 9, and 10]). We shall show at first that "the index of S" [A: S] gives multiplicatively an asymptotic representation of h - as k ranges over infinitely many imaginary abelian fields (for the notation A, see [9] or the following section). every mind matters nhs quizWebStickelberger’s congruences for absolute norms of relative discriminants par Georges GRAS Résumé. Nous généralisons un résultat de J. Martinet sur les congruences de … every mind matters campaign pros and consWeborders to generalize Stickelberger relations [3; 4]. The construction of T(E), which is explained in section 2, works for all self-dual or quasi-Frobenius rings E. The conductor discriminant formula for cyclotomic elds [5, theorem 3.11] ex-presses the discriminant of a cyclotomic ring of integers as a product of conductors. A every mind matters nhs appWebTheorem 1.6 (Stickelberger). We have discZK ≡ 0,1 (mod 4). This theorem is called Stickelberger’s discriminant theorem, among other names. While never stated explicitly in Stickelberger’s work [Sti98], this statement can be deduced from the main results. The modern simple proof given by Schur [Sch29] is typically provided as every mind matters campaign resultsWebAug 12, 2024 · Owen Biesel John Voight Abstract Stickelberger proved that the discriminant of a number field is congruent to 0 or 1 modulo 4. We generalize this to an arbitrary (not necessarily commutative)... brown liquid discharge from rectumWebThe theorem is this: Stickelberger“s Theorem. Let p be an odd prime, f a monk polynomial of degree d with coefficients in ℤ p [ x ], without repeated roots in any splitting field. Let r be … every mind matters loneliness youtube