$p$-adic Valuations and an Introduction to Local Methods — Viewing Integers One Prime at a Time

We define the p-adic valuation v_p(n) and the p-adic absolute value, and state Ostrowski's theorem. We construct the ring of p-adic integers Z_p, prove Hensel's lemma, and discuss the local-global principle.

Folio Official

March 1, 2026

1 The $p$ -adic Valuation

Definition 1 (

p

-adic valuation).

For a prime

p

and a nonzero integer

n

, the $p$ -adic valuation of

n

, denoted

v_{p} (n)

, is the non-negative integer

k

such that

p^{k} ∣ n

and

p^{k + 1} ∤ n

. We set

v_{p} (0) = + \infty

. For a rational number

r = a / b

(

b \neq = 0

), we extend the definition by

v_{p} (r) = v_{p} (a) - v_{p} (b)

Theorem 2 (Properties of the

p

-adic valuation).

For nonzero rationals

x, y

$v_{p} (x y) = v_{p} (x) + v_{p} (y)$ (multiplicativity).
$v_{p} (x + y) \geq min (v_{p} (x), v_{p} (y))$ (ultrametric inequality), with equality whenever $v_{p} (x) \neq = v_{p} (y)$ .

Proof.

(1) Write

x = p^{a} u

and

y = p^{b} v

with

g cd (u, p) = g cd (v, p) = 1

. Then

x y = p^{a + b} uv

with

g cd (uv, p) = 1

, so

v_{p} (x y) = a + b

(2) Let

v_{p} (x) = a

and

v_{p} (y) = b

with

a \leq b

. Then

x + y = p^{a} (u + p^{b - a} v)

, so

v_{p} (x + y) \geq a = min (a, b)

. When

a < b

u + p^{b - a} v \equiv u \neq \equiv 0 (mod p)

, so

v_{p} (x + y) = a

. □

Remark 3.

The fundamental theorem of arithmetic can be restated as follows: every nonzero integer

n

is uniquely expressed as

n = \pm \prod_{p} p^{v_{p} (n)}

, where the product runs over all primes and

v_{p} (n) = 0

for all but finitely many

p

2 The $p$ -adic Absolute Value

Definition 4 (

p

-adic absolute value).

For a nonzero rational

x

, set

∣ x ∣_{p} = p^{- v_{p} (x)}

, and define

∣0 ∣_{p} = 0

. This is called the $p$ -adic absolute value.

Theorem 5 (Properties of the

p

-adic absolute value).

$∣ x ∣_{p} \geq 0$ , with $∣ x ∣_{p} = 0$ if and only if $x = 0$ .
$∣ x y ∣_{p} = ∣ x ∣_{p} ∣ y ∣_{p}$ .
$∣ x + y ∣_{p} \leq max (∣ x ∣_{p}, ∣ y ∣_{p})$ (ultrametric inequality).

Remark 6.

Property (3) is strictly stronger than the ordinary triangle inequality

∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣

. An absolute value satisfying (3) is called non-archimedean.

3 Ostrowski's Theorem

Theorem 7 (Ostrowski's theorem).

Every nontrivial absolute value on

Q

is equivalent either to the usual absolute value

∣ \cdot ∣_{\infty}

or to the

p

-adic absolute value

∣ \cdot ∣_{p}

for some prime

p

Remark 8.

This theorem means that the only ways to complete

Q

are to obtain

R

(the completion with respect to

∣ \cdot ∣_{\infty}

) or

Q_{p}

(the completion with respect to

∣ \cdot ∣_{p}

). The collection of all primes

p

together with

\infty

constitutes the set of places of

Q

, and these places govern the global structure of number theory.

4 The $p$ -adic Integers $Z_{p}$

Definition 9 (

p

-adic integers).

The set of

p

-adic numbers

x \in Q_{p}

with

∣ x ∣_{p} \leq 1

is called the ring of $p$ -adic integers and is denoted

Z_{p}

. Formally, it can be constructed as the inverse limit

Z_{p} = lim Z / p^{n} Z

Z_{p} = {(a_{n})_{n \geq 1} \in n = 1 \prod \infty Z / p^{n} Z a_{n + 1} \equiv a_{n} (mod p^{n}) for all n} .

Theorem 10.

Z_{p}

is an integral domain with fraction field

Q_{p}

. Its unique maximal ideal is

p Z_{p}

, and

Z_{p} / p Z_{p} ≅ Z / p Z

. In other words,

Z_{p}

is a discrete valuation ring.

5 Hensel's Lemma

Theorem 11 (Hensel's lemma).

Let

f (x) \in Z_{p} [x]

and suppose

a \in Z_{p}

satisfies

∣ f (a) ∣_{p} < ∣ f^{'} (a) ∣_{p}^{2}

. Then there exists a unique

α \in Z_{p}

with

f (α) = 0

and

∣ α - a ∣_{p} \leq ∣ f (a) / f^{'} (a) ∣_{p} < ∣ f^{'} (a) ∣_{p}

Proof.

The Newton iteration

a_{n + 1} = a_{n} - f (a_{n}) / f^{'} (a_{n})

converges in the

p

-adic topology. The ultrametric inequality ensures that

∣ a_{n + 1} - a_{n} ∣_{p}

shrinks rapidly and

∣ f (a_{n}) ∣_{p}

decreases at twice the rate (quadratic convergence). By the completeness of

Z_{p}

, the limit

α = lim a_{n}

exists and satisfies

f (α) = 0

. □

Example 12.

Take

p = 7

and

f (x) = x^{2} - 2

. With

a = 3

, we have

f (3) = 7

f^{'} (3) = 6

∣ f (3) ∣_{7} = 1/7 < 1 = ∣ f^{'} (3) ∣_{7}^{2}

. By Hensel's lemma,

2 \in Z_{7}

; that is,

2

has a square root in the

7

-adic integers.

6 The Local–Global Principle

Remark 13.

The local–global principle (or Hasse principle) asserts that an equation has a solution over

Q

if and only if it has solutions over

R

and over

Q_{p}

for every prime

p

. For quadratic forms this is established by the Hasse–Minkowski theorem, but the principle fails in general. A classical counterexample is the equation

3 x^{3} + 4 y^{3} + 5 z^{3} = 0

, which has solutions in

R

and in every

Q_{p}

but not in

Q

Number Theory Algebra Textbook p-adic Numbers Valuation

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

$p$-adic Valuations and an Introduction to Local Methods — Viewing Integers One Prime at a Time

Folio Official

March 1, 2026

1 The $p$ -adic Valuation

Definition 1 (

p

-adic valuation).

For a prime

p

and a nonzero integer

n

, the $p$ -adic valuation of

n

, denoted

v_{p} (n)

, is the non-negative integer

k

such that

p^{k} ∣ n

and

p^{k + 1} ∤ n

. We set

v_{p} (0) = + \infty

. For a rational number

r = a / b

(

b \neq = 0

), we extend the definition by

v_{p} (r) = v_{p} (a) - v_{p} (b)

Theorem 2 (Properties of the

p

-adic valuation).

For nonzero rationals

x, y

$v_{p} (x y) = v_{p} (x) + v_{p} (y)$ (multiplicativity).
$v_{p} (x + y) \geq min (v_{p} (x), v_{p} (y))$ (ultrametric inequality), with equality whenever $v_{p} (x) \neq = v_{p} (y)$ .

Proof.

(1) Write

x = p^{a} u

and

y = p^{b} v

with

g cd (u, p) = g cd (v, p) = 1

. Then

x y = p^{a + b} uv

with

g cd (uv, p) = 1

, so

v_{p} (x y) = a + b

(2) Let

v_{p} (x) = a

and

v_{p} (y) = b

with

a \leq b

. Then

x + y = p^{a} (u + p^{b - a} v)

, so

v_{p} (x + y) \geq a = min (a, b)

. When

a < b

u + p^{b - a} v \equiv u \neq \equiv 0 (mod p)

, so

v_{p} (x + y) = a

. □

Remark 3.

The fundamental theorem of arithmetic can be restated as follows: every nonzero integer

n

is uniquely expressed as

n = \pm \prod_{p} p^{v_{p} (n)}

, where the product runs over all primes and

v_{p} (n) = 0

for all but finitely many

p

2 The $p$ -adic Absolute Value

Definition 4 (

p

-adic absolute value).

For a nonzero rational

x

, set

∣ x ∣_{p} = p^{- v_{p} (x)}

, and define

∣0 ∣_{p} = 0

. This is called the $p$ -adic absolute value.

Theorem 5 (Properties of the

p

-adic absolute value).

$∣ x ∣_{p} \geq 0$ , with $∣ x ∣_{p} = 0$ if and only if $x = 0$ .
$∣ x y ∣_{p} = ∣ x ∣_{p} ∣ y ∣_{p}$ .
$∣ x + y ∣_{p} \leq max (∣ x ∣_{p}, ∣ y ∣_{p})$ (ultrametric inequality).

Remark 6.

Property (3) is strictly stronger than the ordinary triangle inequality

∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣

. An absolute value satisfying (3) is called non-archimedean.

3 Ostrowski's Theorem

Theorem 7 (Ostrowski's theorem).

Every nontrivial absolute value on

Q

is equivalent either to the usual absolute value

∣ \cdot ∣_{\infty}

or to the

p

-adic absolute value

∣ \cdot ∣_{p}

for some prime

p

Remark 8.

This theorem means that the only ways to complete

Q

are to obtain

R

(the completion with respect to

∣ \cdot ∣_{\infty}

) or

Q_{p}

(the completion with respect to

∣ \cdot ∣_{p}

). The collection of all primes

p

together with

\infty

constitutes the set of places of

Q

, and these places govern the global structure of number theory.

4 The $p$ -adic Integers $Z_{p}$

Definition 9 (

p

-adic integers).

The set of

p

-adic numbers

x \in Q_{p}

with

∣ x ∣_{p} \leq 1

is called the ring of $p$ -adic integers and is denoted

Z_{p}

. Formally, it can be constructed as the inverse limit

Z_{p} = lim Z / p^{n} Z

Z_{p} = {(a_{n})_{n \geq 1} \in n = 1 \prod \infty Z / p^{n} Z a_{n + 1} \equiv a_{n} (mod p^{n}) for all n} .

Theorem 10.

Z_{p}

is an integral domain with fraction field

Q_{p}

. Its unique maximal ideal is

p Z_{p}

, and

Z_{p} / p Z_{p} ≅ Z / p Z

. In other words,

Z_{p}

is a discrete valuation ring.

5 Hensel's Lemma

Theorem 11 (Hensel's lemma).

Let

f (x) \in Z_{p} [x]

and suppose

a \in Z_{p}

satisfies

∣ f (a) ∣_{p} < ∣ f^{'} (a) ∣_{p}^{2}

. Then there exists a unique

α \in Z_{p}

with

f (α) = 0

and

∣ α - a ∣_{p} \leq ∣ f (a) / f^{'} (a) ∣_{p} < ∣ f^{'} (a) ∣_{p}

Proof.

The Newton iteration

a_{n + 1} = a_{n} - f (a_{n}) / f^{'} (a_{n})

converges in the

p

-adic topology. The ultrametric inequality ensures that

∣ a_{n + 1} - a_{n} ∣_{p}

shrinks rapidly and

∣ f (a_{n}) ∣_{p}

decreases at twice the rate (quadratic convergence). By the completeness of

Z_{p}

, the limit

α = lim a_{n}

exists and satisfies

f (α) = 0

. □

Example 12.

Take

p = 7

and

f (x) = x^{2} - 2

. With

a = 3

, we have

f (3) = 7

f^{'} (3) = 6

∣ f (3) ∣_{7} = 1/7 < 1 = ∣ f^{'} (3) ∣_{7}^{2}

. By Hensel's lemma,

2 \in Z_{7}

; that is,

2

has a square root in the

7

-adic integers.

6 The Local–Global Principle

Remark 13.

The local–global principle (or Hasse principle) asserts that an equation has a solution over

Q

if and only if it has solutions over

R

and over

Q_{p}

for every prime

p

. For quadratic forms this is established by the Hasse–Minkowski theorem, but the principle fails in general. A classical counterexample is the equation

3 x^{3} + 4 y^{3} + 5 z^{3} = 0

, which has solutions in

R

and in every

Q_{p}

but not in

Q

Number Theory Algebra Textbook p-adic Numbers Valuation

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

$p$-adic Valuations and an Introduction to Local Methods — Viewing Integers One Prime at a Time

1 The $p$ -adic Valuation

2 The $p$ -adic Absolute Value

3 Ostrowski's Theorem

4 The $p$ -adic Integers $Z_{p}$

5 Hensel's Lemma

6 The Local–Global Principle

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