P-Adic Numbers Introduction Cheat Sheet

This cheat sheet introduces $p$ -adic numbers, a number system built by measuring divisibility by a fixed prime $p$ . Students need it because $p$ -adic methods connect number theory, algebra, and modular arithmetic in a precise analytic setting. The main idea is that numbers are close when their difference is divisible by a high power of $p$ .

Key Facts

For a nonzero rational number $x$ , the $p$ -adic valuation $v_p(x)$ is the exponent of $p$ in the factorization of $x$ , with $v_p(0)=\infty$ .
If $x=p^k\frac{a}{b}$ where $p\nmid a$ and $p\nmid b$ , then $v_p(x)=k$ .
The $p$ -adic absolute value is $|x|_p=p^{-v_p(x)}$ for $x\neq 0$ , and $|0|_p=0$ .
The $p$ -adic distance between $x$ and $y$ is $d_p(x,y)=|x-y|_p$ .
The ultrametric inequality says $|x+y|_p\leq \max(|x|_p,|y|_p)$ , which is stronger than the usual triangle inequality.
A sequence $(a_n)$ converges $p$ -adically to $a$ exactly when $v_p(a_n-a)\to \infty$ .
Every $p$ -adic integer has an expansion $a=a_0+a_1p+a_2p^2+\cdots$ with digits $a_i\in\{0,1,\dots,p-1\}$ .
The ring of $p$ -adic integers is $\mathbb{Z}_p=\{x\in\mathbb{Q}_p:|x|_p\leq 1\}$ .

Vocabulary

$p$ -adic valuation: The function $v_p(x)$ that counts the exponent of the prime $p$ in a nonzero rational number $x$ .
$p$ -adic absolute value: The norm $|x|_p=p^{-v_p(x)}$ that makes numbers smaller when they are divisible by larger powers of $p$ .
Ultrametric inequality: The rule $|x+y|_p\leq \max(|x|_p,|y|_p)$ , which changes many geometric intuitions from ordinary real distance.
$p$ -adic integer: An element of $\mathbb{Z}_p$ whose $p$ -adic absolute value satisfies $|x|_p\leq 1$ .
$p$ -adic expansion: A series $a_0+a_1p+a_2p^2+\cdots$ with digits $a_i\in\{0,1,\dots,p-1\}$ that represents a $p$ -adic integer.
$\mathbb{Q}_p$: The field of $p$ -adic numbers, formed by completing $\mathbb{Q}$ using the $p$ -adic distance.

Common Mistakes to Avoid

Using ordinary size to judge $p$ -adic size, which is wrong because $|p^k|_p=p^{-k}$ gets smaller as $k$ increases.
Forgetting that $v_p(xy)=v_p(x)+v_p(y)$ , which leads to incorrect norms because $|xy|_p=|x|_p|y|_p$ .
Treating $p$ -adic expansions like decimal expansions, which is wrong because powers of $p$ grow to the left in real size but become smaller $p$ -adically.
Assuming convergence means terms look close on the real number line, which is wrong because $a_n\to a$ in $\mathbb{Q}_p$ requires $v_p(a_n-a)\to\infty$ .
Applying the ordinary triangle inequality as equality intuition, which is misleading because the stronger bound $|x+y|_p\leq\max(|x|_p,|y|_p)$ often dominates.

Practice Questions

1 For $p=3$ , compute $v_3(81)$ and $|81|_3$ .
2 For $p=5$ , write $\frac{75}{2}$ in the form $5^k\frac{a}{b}$ with $5\nmid a$ and $5\nmid b$ , then find $v_5\left(\frac{75}{2}\right)$ and $\left|\frac{75}{2}\right|_5$ .
3 For $p=2$ , determine whether the sequence $a_n=1+2+2^2+\cdots+2^n$ is Cauchy in the $2$ -adic metric.
4 Explain why two integers can be very far apart in the usual real distance but very close in the $p$ -adic distance.

Understanding P-Adic Numbers Introduction

A useful way to picture p-adic arithmetic is to think in base p, but with infinitely many digits extending toward larger powers of p. Ordinary decimal expansions can continue to the right of the decimal point. P-adic expansions continue in the opposite direction, through powers such as p, p squared, and p cubed.

This makes carrying especially important. When two digits add to at least p, the extra amount is carried into the next higher power.

To find the first several p-adic digits of a sum or product, students can work modulo successively larger powers of p. Each new calculation fixes one more digit without changing the earlier digits.

The strong triangle rule has geometric consequences that feel strange at first. A p-adic ball is a set of numbers sharing a certain number of initial base p digits. Two such balls either do not meet, or one lies completely inside the other.

There is no partial overlap like the overlap of ordinary intervals. Every point inside a ball can serve as its center. This structure makes many arguments cleaner because a small error stays controlled under addition.

There is one important warning. If two numbers have the same p-adic size, their leading digits can cancel.

Their sum may then become much smaller than either original number. Checking for this cancellation is a central habit in calculations.

Completion creates numbers that cannot be written as ordinary rational numbers, in the same way that real numbers complete the rationals in a different sense. A series whose terms contain higher and higher powers of p has a well defined p-adic limit, even if its ordinary size grows without bound. For example, in the three-adic setting, the endless expression two plus three plus nine plus twenty-seven and so on represents one divided by two.

This is correct because multiplying the expression by two gives one after the infinite carrying is interpreted p-adically. More generally, a fraction with denominator not divisible by p has a p-adic expansion. Its digits can be found by solving the fraction modulo p, then modulo p squared, then modulo larger powers.

One major use of this process is lifting solutions of polynomial equations. A solution found modulo a prime can often be improved to a solution modulo the prime squared, then the prime cubed, continuing forever. When the polynomial changes by a nonzero amount near that first solution modulo the prime, the lift is usually unique.

This idea is called Hensel lifting. It turns a difficult infinite problem into a chain of finite modular calculations. Students meet this method in number theory, algebra, coding theory, and proofs about integer solutions.

When learning the topic, keep ordinary size separate from p-adic size. A large power of p is tiny p-adically.

Negative valuation means that p occurs in a denominator. Careful work with residues, carries, and possible cancellation prevents most early mistakes.

Sign in to save

Sign in to save

P-Adic Numbers Introduction Cheat Sheet

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

Key Facts

Vocabulary

Common Mistakes to Avoid

Practice Questions

Understanding P-Adic Numbers Introduction