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Stickelberger's theorem

Webexploiting the classical theorem that the class-group is annihilated by the (Galois-module action of) the so-called Stickelberger ideal. Under some plausible number-theoretical hypothesis, our approach provides a close principal multiple in quantum polynomial time. Combined with thepreviousresults,thissolvesIdeal-SVPintheworstcaseinquantum WebAug 12, 2024 · Stickelberger's discriminant theorem for algebras. Stickelberger proved that the discriminant of a number field is congruent to 0 or 1 modulo 4. We generalize this to …

On the index of the Stickelberger ideal and the ... - ScienceDirect

In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer (1847) while the general result is due to Ludwig Stickelberger (1890). See more Let Km denote the mth cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the mth roots of unity to $${\displaystyle \mathbb {Q} }$$ (where m ≥ 2 is an integer). It is a Galois extension of See more • Gross–Koblitz formula • Herbrand–Ribet theorem • Thaine's theorem • Jacobi sum See more Stickelberger's Theorem Let F be an abelian number field. Then, the Stickelberger ideal of F annihilates the class group of F. Note that θ(F) itself need not be an annihilator, but any multiple of it in Explicitly, the … See more • PlanetMath page See more WebKummer [16] discovered that the Stickelberger ideal S∆ of the group ring Z[∆] annihilates the ideal class group of K. In [7, Theorem 136], Hilbert gave an alternative proof of this important theorem. A new ingredient of his proof is that it uses the theorem of Hilbert and Speiser on the ring of integers of a tame abelian extension over Q ... free intel drivers update https://fredstinson.com

On a generalization of Stickelberger’s Theorem - ResearchGate

WebSTICKELBERGER AND THE EIGENVALUE THEOREM DAVID A. COX To David Eisenbud on the occasion of his 75th birthday. Abstract. This paper explores the relation between the … WebWe prove two versions of Stickelberger s Theorem for positive dimensions and use them to compute the connected and irreducible components of a complex algebraic variety. If the variety is given by polynomials of degree d in n variables, then our ... WebMar 1, 2015 · The theorem states that the Stickelberger element, θ = ∑ a = 1 p − 1 a σ a − 1 ∈ Z [ Gal ( F / Q)] is an annhilator for the class group of F, where p − 1 is the size of Gal ( F / Q). blue chip support

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Stickelberger's theorem

Stickelberger

WebThese remarks and Theorem 3.1 show that the proof of Theorem 1.1 reduces to the following linear algebra setting. Let H be a commutative K -algebra (with unit) given b y some basis b 1 , . . . , b ... WebAs an application, we prove Herbrand’s theorem which relates the nontriviality of certain parts of the ideal class group of ℚ (ζ p ) to p dividing corresponding Bernoulli numbers. …

Stickelberger's theorem

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WebThis theorem tells us about certain elements of the integral group ring of the Galois group of an abelian number field which annihilate the ideal class group of the number field.

WebGrup simetris S n pada himpunan terbatas simbol n adalah grup yang elemennya adalah semua permutasi dari simbol n , dan yang operasi grup merupakan komposisi dari permutasi semacam itu, yang diperlakukan sebagai fungsi bijektiva dari himpunan simbol ke simbol itu sendiri. Karena ada n !(n faktorial) kemungkinan permutasi dari satu set simbol n , maka … http://www.scheiblechner.ch/files/stickelberger.pdf

WebDec 19, 2008 · A non-abelian Stickelberger theorem. David Burns, Henri Johnston. Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring Z_ (p) [G] that annihilates the p-part of the ... WebA non-abelian Stickelberger theorem Note that, in particular, we are in case (i) if Gis isomorphic to the symmetric group on any number of elements, the quaternion group of order eight, or any direct product of such groups. We give the proof of the following corollary in x12, after the proof of Theorem1.2. Corollary 1.5.

WebApr 1, 1985 · Hence, Theorem 2.1 in [9] implies [A: S] -> oo (d--, at least. Further, by the Hermite-Minkowski theorem, we can restate Theorem 1 as follows: For any positive number e, the inequality log [A: S] logh- -1

WebWe note that Zeng (2008) also proved a generalization of Stickelberger’s Theorem for zero-dimensional varieties given by equations and inequations. 1.3. The Hypersurface Case If … bluechip technologies congo rd saWebFeb 9, 2024 · Stickelberger’s theorem Theorem 1 (Stickelberger). Let L= Q(ζm) L = Q ( ζ m) be a cyclotomic field extension of Q Q with Galois group G= {σa}a∈(Z/mZ)× G = { σ a } a ∈ ( Z / m Z) ×, and consider the group ring Q[G] Q [ G]. Define the Stickelberger element θ∈ Q[G] θ ∈ Q [ G] by and take β∈ Z[G] β ∈ Z [ G] such that βθ∈Z[G] β θ ∈ Z [ G] as well. bluechip technologies asia sri lankaWebSTICKELBERGER’S DISCRIMINANT THEOREM FOR ALGEBRAS ASHER AUEL, OWEN BIESEL, AND JOHN VOIGHT Abstract. Stickelberger proved that the discriminant of a number eld … free intelligence courses onlineWebThe theorem follows by proving that ΔK ≡ 0 mod 2 implies ΔK ≡ 0 mod 4, and that ΔK ≡ 1 mod 2 implies ΔK ≡ 1 mod 4. Assume that 2 ΔK. Then there exists a prime p of K lying … free intelligence gamesWebThe aim of this chapter is to give, for any abelian number field, elements of the group ring of the Galois group which annihilate the ideal class group.hey will form the Stickelberger … free intelligence analyst trainingWebStatements of the Theorem Let t et−1 = P∞ n=0 Bk tn n!. Bn is called Bernoulli numbers. A fact states that (1 n) = Bn n for n 1. In the 1930s, Herbrand found: Proposition (Herbrand,1930s) Let k 2 [2;p 3] be an even integer. If C(˜1−k) 6= 0, then pjBk. This is a consequence of the Stickelberger’s Theorem. bluechip technologies limitedhttp://www.numdam.org/item/CM_1985__55_2_209_0.pdf bluechip technologies ltd