Chapter 5 Definitions

Question 1

Z_p

Answer 1

Z_p is a ring from p-adic integers for each prime P.

Z ⊂ Z_p

Question 2

Field of Fractions

Answer 2

Z_p has a field of fractions (Q_p

(Q ⊂ (Q_p

(Q_p = { a/b | a,b ϵ Z_p, b≠0 }

Question 3

A p-adic integer

Answer 3

Fix a prime number P.

A P-adic integer α is defined by a sequence of integers x_k for k ≥ 1

α = { x_k }_k=1^∞ = {x₁, x₂, … } satisfying:

x_k+1 ≡ x_k (mod p^k ) for all k ≥ 1

with two sequences {x_k} and {y_k} determining the same p-adic integer α ⇔ x_k ≡ y_k (mod p^k) for all k ≥ 1.

The set of all p-adic integers is denoted Z_p.

Question 4

Coherant

Answer 4

An integer satisfying x_k+1 ≡ x_k (mod p^k ) for all k ≥ 1 is called coherant.

Question 5

Reduced representation of a p-adic integer

Answer 5

The repesentation of a p-adic integer x={x_k} is called reduced if 0 ≤ x_k < p^k for all k ≥ 1.

Question 6

Rational integers

Answer 6

We call elements of Z rational integers to distinguish them from p-adic integers.

Question 7

Set of p-integral rationals

Answer 7

The set of p-integral rationals is:

R_p = { n/d ϵ (Q : p ∤ d }

= { x ϵ (Q : ord_p(x) ≥ 0 }

Question 8

ord_p(α) for α ϵ Z_p

Answer 8

For non-zero α ϵ Z_p, ordp(α) = m where m is the largest integer for which p^m | α (in Z_p).

ord_p(0) = ∞

Question 9

Norm on a field F

Answer 9

A norm on a field F is a function x →‖x‖ s.t:

1) ‖x‖ ≥ 0 and ‖x‖ = 0 ⇔ x=0
2) ‖x y‖ = ‖x‖ ‖y‖
3) ‖ x + y ‖ ≤ ‖x‖ + ‖y‖

Question 10

Trivial Norm

Answer 10

‖x‖ = 1 for all non-zero x

Question 11

Metric

Answer 11

d(x,y) = ‖ x - y ‖ for all x, y ϵ F

1) d(x,y) ≥ 0 and d(x,y) = 0 ⇔ x = y
2) d(x,y) = d(y,x)
3) d(x,y) ≤ d(x,z) + d(z,y)

Question 12

‖ · ‖ is equivalent to ‖ · ‖^a for all a in |R⁺

Question 13

P-adic norm

Answer 13

Let p be a prime.

For non-zero x ϵ (Q_p the p-adic norm of x is

|x|_p = p^-ord_p(x) .

|0|_p = 0