Download PDF by Gove W. Effinger: Additive Number Theory of Polynomials Over a Finite Field

By Gove W. Effinger

ISBN-10: 019853583X

ISBN-13: 9780198535836

This quantity is a scientific remedy of the additive quantity concept of polynomials over a finite box, a space owning deep and engaging parallels with classical quantity thought. In offering asymptomatic proofs of either the Polynomial 3 Primes challenge (an analog of Vinogradov's theorem) and the Polynomial Waring challenge, the booklet develops a number of the instruments essential to practice an adelic "circle approach" to a wide selection of additive difficulties in either the polynomial and classical settings. A key to the tools hired this is that the generalized Riemann speculation is legitimate during this polynomial atmosphere. The authors presuppose a familiarity with algebra and quantity idea as should be received from the 1st years of graduate direction, yet another way the publication is self-contained. beginning with research on neighborhood fields, the most technical effects are all proved intimately in order that there are huge discussions of the idea of characters in a non-Archimidean box, adele classification teams, the worldwide singular sequence and Radon-Nikodyn derivatives, L-functions of Dirichlet variety, and K-ideles.

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4. Satz (Fermat). Sei p eine Primzahl. Dann gilt f¨ ur jede nicht durch p teilbare ganze Zahl a ap−1 ≡ 1 mod p. Beweis. Wir k¨ onnen a, genauer a mod p, als Element der multiplikativen Gruppe (Z/pZ)∗ = F∗p auffassen, die aus p − 1 Elementen besteht. 3. Bemerkung. 4 wird manchmal als der kleine Satz von Fermat bezeichnet. Als den großen Satz von Fermat bezeichnet man die Behauptung von Fermat, dass die Gleichung xn + y n = z n f¨ ur n 3 keine ganzzahligen L¨osungen mit xyz = 0 besitzt. Dies war 300 Jahre lang nur eine Vermutung, bevor diese Behauptung 1995 von Andrew Wiles [wiles] bewiesen werden konnte.

Da R Hauptidealring ist, gibt es ein d ∈ R mit (a, x) = (d). Nat¨ urlich ist d = 0. Falls d ∈ R∗ , folgt aus der Irreduzibilit¨at von a, dass d und a assoziiert sind, also d | x ⇒ a | x, entgegen der Voraussetzung. Also ist d ∈ R∗ , es gibt also μ, ν ∈ R mit μa + νx = 1. Daher ist y = μay + νxy = μay + νaq. d. Insbesondere fallen also im Ring Z der ganzen Zahlen die Begriffe irreduzibel und prim zusammen. Wir werden unter Primzahlen in Z immer die positiven Primelemente von Z verstehen. S¨amtliche Primelemente von Z haben die Gestalt ±p, wobei p die Menge der Primzahlen {2, 3, 5, 7, 11, .

Definition. Ein Polynom n ai X i ∈ Z[X] F (X) = i=0 heißt primitiv, wenn der gr¨oßte gemeinsame Teiler seiner Koeffizienten ai gleich 1 ist. § 5 Primfaktor-Zerlegung 40 Ist ein Polynom aus Z[X] nicht primitiv, so kann man den gr¨oßten gemeinsamen Teiler der Koeffizienten ausklammern. Es ergibt sich also, dass jedes Polynom aus Q[X] zu einem primitiven Polynom aus Z[X] assoziiert ist. Dieses primitive Polynom ist bis aufs Vorzeichen eindeutig bestimmt. 8. Lemma. Das Produkt zweier primitiver Polynome F, G ∈ Z[X] ist wieder primitiv.

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Additive Number Theory of Polynomials Over a Finite Field by Gove W. Effinger

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