• Aucun résultat trouvé

Show that lcm{1,2


Academic year: 2022

Partager "Show that lcm{1,2"


Texte intégral


MATH-GA2210.001 : Final Exam, May, 11, 2016 Lecture notes are authorized.

The exam is posted online on May, 11, 2016, by 7 pm.

The copies should be sent by mail (either a scan, or a pdf file) to alena.pirutka@gmail.com

The deadline to send the corrections is May, 12, 2016, 7pm.

1. (a) Recall that

ψ(x) = X



Show that

lcm{1,2, . . . , n}=eψ(n),

where the left hand side is the least commun multiple of the integers 1,2, . . . , n.

(b) Show that

eψ(2n+1) Z 1



is a positive integer. Deduce that

ψ(2n+ 1)≥2n log2.

2. (a) Let K = Qpp), where ζp is a primitive pth root of unity. Show that [K :Qp] =p−1 and that this extension is totally ramified.

(b) (*) Which quadratic extensions of Q2 are ramified ? 3. Let K =Q(√

−31). Let α= 1+


2 .

(a) DetermineOK and disc(OK).

(b) Determine the ideals ofOK containing 2or3. Among these ideals, which are principal (resp. prime) ?

(c) Decompose the ideals (2), (3), and (α) as a product of prime ideals in OK.

(d) Determine the groupCl(OK).


Documents relatifs

Thanks to Theorem 1.1, for any Kantorovich potential u and any optimal transport plan g, u is decreasing at maximal slope on any maximal transport ray; this implies, in particular,

Calculate the probability such that there is exactly one goal during the first half condi- tionally upon the fact that “les girondins” scored two goals during the football match..

By Baire’s theorem, an infinite dimensional Fréchet space cannot have a countable algebraic

M2R Université Grenoble Alpes 2019-2020 Introduction to analytic geometry (course by Jean-Pierre Demailly).. Sheet number 7, 21/11/2019 The goal of this sheet is to detail the proof

Derive from there that the Hodge decomposition is independent of the choice of the Kähler metric

Dans le scénario illustré par la figure 28, les nœuds ne sont pas mobiles mais sont soumis à des interférences. Après une certaine intensité du bruit, les

Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale.. Toute copie ou impression de ce fichier doit contenir la présente mention

Ces charges, qui grèvent un fonds au profit d'un autre fonds, sont de trois sortes : certaines permettent de faire passer les canalisations d’eau sur les terres d’autrui

[j -invariant] Let k be an algebraically closed field of characteristic different from 2

Also use the formulas for det and Trace from the



Can −1 be a quadratic residue in some congruence

.} be the set of positive integers which have the last digit 1 when written in base 10.. Determine all Dirichlet characters modulo 8 and



.} be the set of positive integers which have the last digit 1 when written in



Universit´ e Paris Diderot Morningside Center for Mathematics.. Exercises (for beginners) on

We define a partition of the set of integers k in the range [1, m−1] prime to m into two or three subsets, where one subset consists of those integers k which are < m/2,

Drawing together a diverse range of artistic practices including film, video, photography, sculpture, text installation, performance, and mixed media, the works presented in Among

In 1980, artist, curator and writer John Watt produced, curated and contributed to Television by Artists, a series of video art that was aired on Rogers Cable TV between M ay