# An Introduction to p-adic Numbers and p-adic Analysis by Andrew Baker

By Andrew Baker

Similar group theory books

Semigroup theory and evolution equations: the second international conference

Court cases of the second one foreign convention on traits in Semigroup concept and Evolution Equations held Sept. 1989, Delft collage of know-how, the Netherlands. Papers care for fresh advancements in semigroup idea (e. g. , confident, twin, integrated), and nonlinear evolution equations (e

Topics in Galois Theory

Written through one of many significant participants to the sector, this ebook is full of examples, routines, and open difficulties for extra edification in this interesting subject.

Products of Finite Groups (De Gruyter Expositions in Mathematics)

The research of finite teams factorised as a made of or extra subgroups has turn into an issue of serious curiosity over the last years with purposes not just in crew thought, but additionally in different components like cryptography and coding idea. It has skilled a huge impulse with the advent of a few permutability stipulations.

Automorphic Representation of Unitary Groups in Three Variables

The aim of this e-book is to strengthen the good hint formulation for unitary teams in 3 variables. The sturdy hint formulation is then utilized to procure a class of automorphic representations. This paintings represents the 1st case within which the strong hint formulation has been labored out past the case of SL (2) and similar teams.

Additional info for An Introduction to p-adic Numbers and p-adic Analysis [Lecture notes]

Sample text

The function ω : Zp −→ Qp is locally constant and satisﬁes the conditions ω(αβ) = ω(α)ω(β), |ω(α + β) − ω(α) − ω(β)|p < 1. Moreover, the image of this function consists of exactly p elements of Zp , namely the p distinct roots of the polynomial X p − X. Proof. The multiplicative part follows from the deﬁnition, while the additive result is an easy exercise with the ultrametric inequality. For the image of ω, we remark that the distinct numbers in the list 0, 1, 2, . . , p − 1 satisfy |r − s|p = 1.

The open disc centred at α of radius δ is D (α; δ) = {γ ∈ Qp : |γ − α|p < δ}. The closed disc centred at α of radius δ is D (α; δ) = {γ ∈ Qp : |γ − α|p δ}. Clearly D (α; δ) ⊆ D (α; δ). Such a notion is familiar in the real or complex numbers; however, here there is an odd twist. 2. Let β ∈ D (α; δ). Then D (β; δ) = D (α; δ) . Hence every element of D (α; δ) is a centre. Similarly, if β ′ ∈ D (α; δ), then D (β ′ ; δ) = D (α; δ). Proof. This is a consequence of the fact that the p-adic norm is non-Archimedean.

M. Robert, A course in p-adic analysis, Springer-Verlag, 2000. 53 Problems Problem Set 1 1-1. For each of the following values n = 19, 27, 60, in the ring Z/n ﬁnd (i) all the zero divisors, (ii) all the units and their inverses. 1-2. Let f (X) = X 2 − 2 ∈ Z[X]. For each of the primes p = 2, 3, 7, determine whether or not there is a root of f (X): (i) mod p, (ii) mod p2 , mod p3 , (iii) (iv) mod p4 . Can you say anything more? 1-3. Solve the following system of simultaneous linear equations over Z/n for each of the values n = 2, 9, 10: 3x + 2y − 11z ≡ n 7x + ≡ 12 2z − 8y + 1 n z ≡ n 2 1-4.